so we have to start preparing for the trig exam
so away we go
**The Unit Circle
90 degrees, (0,1), pi/2
180 degrees, (-1,0), pi
270 degrees, (0,-1), 3pi/2
360 degrees, (1,0), 2pi
**6 Trig Functions
sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y
**Degrees & Radians
Degrees to radians= Degree * pi/180
Radians to degrees= Radians * 180/pi
**To solve coterminal angles, either add or subtract 360 to the angle
i need a review of the chapter with amplitude and the graphs and other information that goes along with a problem like that
Tuesday, March 30, 2010
Stephen's Reflection
Ok so this week i think we are learning something new..idk but anyway i do remember trig functions really well because they are really east. there are six trig functions:
6 Trig Functions
sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y
theres a few things i do not understand which is law of sines, law of cosines, and sigma notations...
6 Trig Functions
sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y
theres a few things i do not understand which is law of sines, law of cosines, and sigma notations...
Sunday, March 28, 2010
Lima Beans, Hearts, and Roses!
- Limacon - Looks like a lima bean!
- r = a+b sin theta
- r = a+b cos theta
- Cardioid - Looks like a heart!
- a-b sin theta
- a-b cos theta
- Rose - Looks like a ...rose.
- r = a sin (number of petals) theta
- r = a cos (number of petals) theta
- Archimedes Spiral - The black and white spiral that hypnotizes people in the cartoons.
- r = a theta +b
- Logarithmic Spiral - Looks like a ...spiral.
- r=a^theta b
Alicia's Reflection #32
Alrighty so last week we finished taking our chapter tests. I think this week is ACT prep so hopefully we review ACT stuff in math. I am going to review some trig from Ch. 10.
Law of Sines:
sinA/a = sinB/b = sinC/c
Law of Cosines:
(opp leg)^2 = (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg)cos(angle between)
Example:
x= 6^2 + 5^2 -2(5)(6) cos 36
x=3.530
Here are some formulas:
Cos(α +/- β)=cos α cos β -/+ sin α sin β
sin(α +/- β)=sin α cos β -/+ cos α sin β
sin x + sin y= 2 sin x + y/2 cos x-y/2
sin x - sin y= 2 cos x + y/2 sin x-y/2
cos x + cos y= 2 cos x + y/2 cos x-y/2
cos x - cos y= 2 sin x + y/2 sin x-y/2
tan (α + β)=tan α + tan β/1-tan α tan β
tan (α - β)=tan α - tan β/1+tan α tan β
sin2α=2sin α cos α
cos 2α=cos^2 α –sin^2 α = 1-2 sin^2 α= 2 cos^2 α -1
tan 2α = 2tan α /1-tan^2 α
sin α/2= +/- √1-cos α/2
cos α/2= +/- √1+ cos α/2
tan α/2= +/- √1-cos α or 1 + cos α
=sin α/1+cos α
=1-cos α/sin α
I could use some help with sigma notation
Law of Sines:
sinA/a = sinB/b = sinC/c
Law of Cosines:
(opp leg)^2 = (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg)cos(angle between)
Example:
x= 6^2 + 5^2 -2(5)(6) cos 36
x=3.530
Here are some formulas:
Cos(α +/- β)=cos α cos β -/+ sin α sin β
sin(α +/- β)=sin α cos β -/+ cos α sin β
sin x + sin y= 2 sin x + y/2 cos x-y/2
sin x - sin y= 2 cos x + y/2 sin x-y/2
cos x + cos y= 2 cos x + y/2 cos x-y/2
cos x - cos y= 2 sin x + y/2 sin x-y/2
tan (α + β)=tan α + tan β/1-tan α tan β
tan (α - β)=tan α - tan β/1+tan α tan β
sin2α=2sin α cos α
cos 2α=cos^2 α –sin^2 α = 1-2 sin^2 α= 2 cos^2 α -1
tan 2α = 2tan α /1-tan^2 α
sin α/2= +/- √1-cos α/2
cos α/2= +/- √1+ cos α/2
tan α/2= +/- √1-cos α or 1 + cos α
=sin α/1+cos α
=1-cos α/sin α
I could use some help with sigma notation
Reflection
So I'm thinking that since B-Rob is suppose to be coming back this week and originally was suppose to be coming back last week we will have A LOTTTT of work to do in order to catch up. I guess I could use it because I really do not remember too much...area of a non right triangle.
Area of a Non Right Triangle:
Formula:1/2(leg)(leg)sin(angle b/w)
So if you had 1/2(3)(6)sin(52) your answer would be: 7.100
Does anyone know what we're suppose to be doing when B-Rob comes back?
Area of a Non Right Triangle:
Formula:1/2(leg)(leg)sin(angle b/w)
So if you had 1/2(3)(6)sin(52) your answer would be: 7.100
Does anyone know what we're suppose to be doing when B-Rob comes back?
Amy's Reflection #32
**Logs
Condense:
Ex) logm + log7 + 4logn
= log7mn^4
Ex) 5loga + logd + log6
= log6da^5
Ex) 4logt - logc
= t^4/c
Ex) logn - 3logh -logy
= n/yh^3
Expand:
Ex) log5gh^2
= log5 + 2logh +logg
Ex) m^3b^7/f
= 3logm + 7logb - logf
**The Unit Circle
90 degrees, (0,1), pi/2
180 degrees, (-1,0), pi
270 degrees, (0,-1), 3pi/2
360 degrees, (1,0), 2pi
**6 Trig Functions
sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y
**Degrees & Radians
Degrees to radians= Degree * pi/180
Radians to degrees= Radians * 180/pi
**To solve coterminal angles, either add or subtract 360 to the angle.
can someone help me with how to use law of cosine??
Condense:
Ex) logm + log7 + 4logn
= log7mn^4
Ex) 5loga + logd + log6
= log6da^5
Ex) 4logt - logc
= t^4/c
Ex) logn - 3logh -logy
= n/yh^3
Expand:
Ex) log5gh^2
= log5 + 2logh +logg
Ex) m^3b^7/f
= 3logm + 7logb - logf
**The Unit Circle
90 degrees, (0,1), pi/2
180 degrees, (-1,0), pi
270 degrees, (0,-1), 3pi/2
360 degrees, (1,0), 2pi
**6 Trig Functions
sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y
**Degrees & Radians
Degrees to radians= Degree * pi/180
Radians to degrees= Radians * 180/pi
**To solve coterminal angles, either add or subtract 360 to the angle.
can someone help me with how to use law of cosine??
Stephanie's Reflection
Limacon
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Tuesday, March 23, 2010
taylors reflection for 21 march
there are two types of limit equations
the ones that use rules and the ones that use a calculator
the ones that use rules have simple hints to memorize for solving
the only ones that use rules are the polynomial equations problems
memorize this
((the rules))
t- top lead co
b- bottom lead co
t=b then coefficients
t>b then infinity
t
if you get a problem with a limit that is a polynomial equation
use the rules.
each and every time
the other type of problem is the one that calls for the use of a calculator
every single problem with limits that is not a polynommial equation calls for the use of a calculator
all you have to do is plug in for n three different times with
100
1000
10000
then plug into calculator
record what each outcome is and decipher what the numbers are headed toward which will then be your answer
what i need help on is memorizing the formulas for chapter eight
they are the only part of trig that i do not understand and since we are only going to have two review weeks before the trig exam i really could used some help...
i dont know if its chapter eight or chapter ten
but i need help on memorizing the formulas from the chapter in trig where you are substituting
the ones that use rules and the ones that use a calculator
the ones that use rules have simple hints to memorize for solving
the only ones that use rules are the polynomial equations problems
memorize this
((the rules))
t- top lead co
b- bottom lead co
t=b then coefficients
t>b then infinity
t
if you get a problem with a limit that is a polynomial equation
use the rules.
each and every time
the other type of problem is the one that calls for the use of a calculator
every single problem with limits that is not a polynommial equation calls for the use of a calculator
all you have to do is plug in for n three different times with
100
1000
10000
then plug into calculator
record what each outcome is and decipher what the numbers are headed toward which will then be your answer
what i need help on is memorizing the formulas for chapter eight
they are the only part of trig that i do not understand and since we are only going to have two review weeks before the trig exam i really could used some help...
i dont know if its chapter eight or chapter ten
but i need help on memorizing the formulas from the chapter in trig where you are substituting
Monday, March 22, 2010
Stephen's Reflection
Ok so this week we dont have a teacher again so theres stuff i need help on. Anyway i really understand and remember trig stuff and trig functions. I understand the 6 trig functions and how to use them.
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
Waht i need help on is the formulas for hyperbolas and circles and stuff like that and i needa know what i need to find for each
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
Waht i need help on is the formulas for hyperbolas and circles and stuff like that and i needa know what i need to find for each
Sunday, March 21, 2010
alaina's blog, 21 march 2010
law of sins :)
sinA/a=sinB/b=sinC/c
*only used when you have pairs, an angle and the side opposite of it.
*setting up a proportion.
Ex: a civil engineer wants to determine the distance from points A and B to an inaccessable point C. from direct measurement -- AB=25m,
first you would draw a diagram and lable EVERYTHING. Then, choose your pairs. Finally set up a proportion.
(sin50/25)=(sin20/B)
cross multiply--Bsin50=25sin20
divide by sin50
B=11.162m
you would follow the same process to find side "a".
I still don't understand integral coefficients if anyone wants to help.
sinA/a=sinB/b=sinC/c
*only used when you have pairs, an angle and the side opposite of it.
*setting up a proportion.
Ex: a civil engineer wants to determine the distance from points A and B to an inaccessable point C. from direct measurement -- AB=25m,
first you would draw a diagram and lable EVERYTHING. Then, choose your pairs. Finally set up a proportion.
(sin50/25)=(sin20/B)
cross multiply--Bsin50=25sin20
divide by sin50
B=11.162m
you would follow the same process to find side "a".
I still don't understand integral coefficients if anyone wants to help.
Reflection
SOHCAHTOA:
sin theta=opposite/hypotenuse
cos theta=adjacent/hypotenuse
tan theta=opposite/adjacent
SOHCAHTOA is used when either you have two sides of a right triangle and you need to find an angle or you have an angle and one side. Here's an example:
A right triangle has 3 angles: 90°, 30°, and 60°. The hypotenuse is x cm. The side opposite the 60° angle is 8 cm. What is the length of the hypotenuse?
You would use the sin formula and the equation would be sin(60)=8/x.
Then you would get .8660=8/x
You divide 8 by .8660 and get 9.2380
So the hypotenuse of the triangle would be 9.2380 cm.
So what exactly are we doing this week in this class?
sin theta=opposite/hypotenuse
cos theta=adjacent/hypotenuse
tan theta=opposite/adjacent
SOHCAHTOA is used when either you have two sides of a right triangle and you need to find an angle or you have an angle and one side. Here's an example:
A right triangle has 3 angles: 90°, 30°, and 60°. The hypotenuse is x cm. The side opposite the 60° angle is 8 cm. What is the length of the hypotenuse?
You would use the sin formula and the equation would be sin(60)=8/x.
Then you would get .8660=8/x
You divide 8 by .8660 and get 9.2380
So the hypotenuse of the triangle would be 9.2380 cm.
So what exactly are we doing this week in this class?
Alicia's Reflection #31
Alrighty so we took our exam on flatland which was easy for the most part. I am going to review some material on circles because I kind of forgot this chapter.
This is the standard form of a circle: (x-h)^2+(y-k)^2=r^2
The center of a circle is: (h,k)
The radiusis represented by: r
*Find the center and radius of the circle.
1.) (x-3)^2+(y+7)^2=19
center: (3,-7)
radius: squareroot of 19
*Find the intersection of the circle.
1.) x^2+4^2-25 and y=2x-2
a) y=2x-2
b) x^2=(2x-2)^2=25
c) x^2+4x^2-8x+4=25
5x^2-8x+4=25
5x^2-8x-21=0
5x^2-15x+7x-21=0
5x(x-3)+7(x-3)=0
(x-3)(5x+7)
x=3 x=-7/5
y=2(x)-2
2(3)-2=4
y=2(-7/5)-2=-24/5
(3,4) (-7/5,-24/5)
*Write in Standard form.
1.) Center: (4,3)
Radius: 2
(x-4)^2+(y-3)^2=4
This is the standard form of a circle: (x-h)^2+(y-k)^2=r^2
The center of a circle is: (h,k)
The radiusis represented by: r
*Find the center and radius of the circle.
1.) (x-3)^2+(y+7)^2=19
center: (3,-7)
radius: squareroot of 19
*Find the intersection of the circle.
1.) x^2+4^2-25 and y=2x-2
a) y=2x-2
b) x^2=(2x-2)^2=25
c) x^2+4x^2-8x+4=25
5x^2-8x+4=25
5x^2-8x-21=0
5x^2-15x+7x-21=0
5x(x-3)+7(x-3)=0
(x-3)(5x+7)
x=3 x=-7/5
y=2(x)-2
2(3)-2=4
y=2(-7/5)-2=-24/5
(3,4) (-7/5,-24/5)
*Write in Standard form.
1.) Center: (4,3)
Radius: 2
(x-4)^2+(y-3)^2=4
Stephanie's Reflection
Sine and Cosine Sum/Difference Formulas:
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
Tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
Tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
Saturday, March 20, 2010
Amy's Reflection #31
here's some stuff from chapter 11..
Imaginary Numbers are no longer "imaginary"
Rectangular form: a + bi
Polar form: z = r cos theta + r sin theta i (abbreviated z = r cis theta)
Examples:
1. Express 2 cis 50degrees in rectangular form
2 cos 50 + 2 sin 50 i
2. Express -1-2i in polar form
radius = +- sqrt of ((-1)^2 + (-2)^2)) = +- sqrt of (5)
theta = tan^-1(-2/-1)
theta = tan^-1(1)
*tangent is positive in the first and third quadrants, 63.435 and 243.435
*63 is positive for cosine so it goes with the positive sqrt of 5
*243 is negative for cosine so it goes with the negative sqrt of 5
z= sqrt of 5 cis 63.435
z= sqrt of 5 cos 63.435 + sqrt of 5 sin 63.435 i
z= negative sqrt of 5 cis 243.435
z= negative sqrt of 5 cos 243.435 + negative sqrt of 5 sin 243.435 i
De Moivre's Theorem: z^n = r^n cis(n)(theta)
Examples:
1. z=2cis20degrees Find z^2
z^2=2^2cis2(20degrees)
z^2=4cis40degrees
2. 4cis15degrees Find z^4
z^4=4^4cis4(15degrees)
z^4=256cis60degrees
Limacon
r=a+b sin(theta)
r=a+b cos(theta)
Cardioid
a-b sin(theta)
r=a-b cos(theta)
Rose
r=a sin(n theta)
r=a cos (n theta)
*n=how many petals
Archimedes Spiral
r=a theta+b
Logarithmic Spiral
r=a b^theta
Examples:
1. r=theta+2
2. r=2+3cos(theta)
3. r=5
4. r=3sin(4 theta)
5. r=1/2(3^theta)
6. r=2sin(theta)
1. archimedes spiral
2. limacon
3. circle with its center at the pole
4. rose with 4 petals
5. logarithmic spiral
6. circle that intersects with the pole
ok what i really dont understand is the first two sections..if someone could explain them to me that would be awesome..thanks..
Imaginary Numbers are no longer "imaginary"
Rectangular form: a + bi
Polar form: z = r cos theta + r sin theta i (abbreviated z = r cis theta)
Examples:
1. Express 2 cis 50degrees in rectangular form
2 cos 50 + 2 sin 50 i
2. Express -1-2i in polar form
radius = +- sqrt of ((-1)^2 + (-2)^2)) = +- sqrt of (5)
theta = tan^-1(-2/-1)
theta = tan^-1(1)
*tangent is positive in the first and third quadrants, 63.435 and 243.435
*63 is positive for cosine so it goes with the positive sqrt of 5
*243 is negative for cosine so it goes with the negative sqrt of 5
z= sqrt of 5 cis 63.435
z= sqrt of 5 cos 63.435 + sqrt of 5 sin 63.435 i
z= negative sqrt of 5 cis 243.435
z= negative sqrt of 5 cos 243.435 + negative sqrt of 5 sin 243.435 i
De Moivre's Theorem: z^n = r^n cis(n)(theta)
Examples:
1. z=2cis20degrees Find z^2
z^2=2^2cis2(20degrees)
z^2=4cis40degrees
2. 4cis15degrees Find z^4
z^4=4^4cis4(15degrees)
z^4=256cis60degrees
Limacon
r=a+b sin(theta)
r=a+b cos(theta)
Cardioid
a-b sin(theta)
r=a-b cos(theta)
Rose
r=a sin(n theta)
r=a cos (n theta)
*n=how many petals
Archimedes Spiral
r=a theta+b
Logarithmic Spiral
r=a b^theta
Examples:
1. r=theta+2
2. r=2+3cos(theta)
3. r=5
4. r=3sin(4 theta)
5. r=1/2(3^theta)
6. r=2sin(theta)
1. archimedes spiral
2. limacon
3. circle with its center at the pole
4. rose with 4 petals
5. logarithmic spiral
6. circle that intersects with the pole
ok what i really dont understand is the first two sections..if someone could explain them to me that would be awesome..thanks..
Thursday, March 18, 2010
Cemments
Q. can someone help me with the trig chart?
A.Trig Chart:
0°
sin0=0
cos0=1
tan0=0
csc0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Q.I dont seem to remember the trig functions...can someone remind me what they are?
A.oh yeah sure man i got you...
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
A.Trig Chart:
0°
sin0=0
cos0=1
tan0=0
csc0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Q.I dont seem to remember the trig functions...can someone remind me what they are?
A.oh yeah sure man i got you...
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
taylors 17 March "comment" blog b/c there were no questions
Im going to post some more review stuff since there were no comments
this is from around chapter nine
Area of a non-right triangle
A=1/2(leg)(leg)sin(angle between)
Area of Right Triangles
A=1/2bh
SOHCAHTOA
sinΘ=opposite leg/hypotenuse
cosΘ=adjacent leg/hypotenuse
tanΘ=opposite leg/adjacent leg
Law of Sines
sinA/a - sinB/b = sinC/c
(used when you know pairs or opposites in a non-right triangle)
Law of Cosines
(opposite leg)²=(adjacent leg)² + (other leg)² - 2(adjacent leg)(adjacent leg)cos°
Area of Inscribed Shapes
A=nr²sinΘcosΘ
this is from around chapter nine
Area of a non-right triangle
A=1/2(leg)(leg)sin(angle between)
Area of Right Triangles
A=1/2bh
SOHCAHTOA
sinΘ=opposite leg/hypotenuse
cosΘ=adjacent leg/hypotenuse
tanΘ=opposite leg/adjacent leg
Law of Sines
sinA/a - sinB/b = sinC/c
(used when you know pairs or opposites in a non-right triangle)
Law of Cosines
(opposite leg)²=(adjacent leg)² + (other leg)² - 2(adjacent leg)(adjacent leg)cos°
Area of Inscribed Shapes
A=nr²sinΘcosΘ
Wednesday, March 17, 2010
Alrighty so we have our exam this week on Flatland. Basically just study the questions we did in class along with the vocabulary words. okay I am going to review some trig laws:
*Law of Sines: sin(opp. angle)/Leg =sin(Opp. angle)/Leg.
Example: triangle ABC where A=36 degrees a=3 and B=56 degrees. find b
sin36/3=sin56/x 3sin56/sin36= x
*Law of Cosines: (opposite leg)^2=(adjacent leg)^2+(other opposite leg)^2-2(leg)(leg)cos(angle in between)
Example: for a triangle with C=36 degrees a=5 b=6
c^2=5^2+6^2-2(5)(6)cos36
c=Square root of(25+36-2(5)(6)cos36)
c= 3.53
*The area of non-right triangle:
1/2(leg)(leg)sin(angle between)
Example: Triangle ABC has sides a=5 b=3 and C=40 degrees
= (1/2)(5)(3)(sin(40))
Goodluck on the exam!!!! :)
*Law of Sines: sin(opp. angle)/Leg =sin(Opp. angle)/Leg.
Example: triangle ABC where A=36 degrees a=3 and B=56 degrees. find b
sin36/3=sin56/x 3sin56/sin36= x
*Law of Cosines: (opposite leg)^2=(adjacent leg)^2+(other opposite leg)^2-2(leg)(leg)cos(angle in between)
Example: for a triangle with C=36 degrees a=5 b=6
c^2=5^2+6^2-2(5)(6)cos36
c=Square root of(25+36-2(5)(6)cos36)
c= 3.53
*The area of non-right triangle:
1/2(leg)(leg)sin(angle between)
Example: Triangle ABC has sides a=5 b=3 and C=40 degrees
= (1/2)(5)(3)(sin(40))
Goodluck on the exam!!!! :)
Tuesday, March 16, 2010
taylor 15 march reflection
Since we are focusing on review tests right now i figured id start reviewing on the more recent chapters because those will start becomming foggy to us as we review the early chapters.
Graph shapes and their formulas
Limacon
r=a+b sin(theta)
r=a+b cos(theta)
Cardioid
a-b sin(theta)
a-b cos(theta)
Rose
r=a sin(n theta)
r=a cos (n theta)
(n=how many petals {if n isodd[#=n] if n is even [#=2n]}
Archimedes Spiral
r=a theta+b
Logarithmic Spiral
r=a^theta b
CONVERTING
when going from polar to rectangular you plug into
X=rcos(theta)
Y=rsin(theta)
and work out until you get a x point and a y point
when going from rectangular to polar you plug into
r=+/- squareroot X^2 +Y^2
and
Theta= (Y/X)
once youve solved for both of these you"ll plug into (+r, theta) (-r, theta)
i need help with a review of the formulas from the triangle section if anyone can help please do!
Graph shapes and their formulas
Limacon
r=a+b sin(theta)
r=a+b cos(theta)
Cardioid
a-b sin(theta)
a-b cos(theta)
Rose
r=a sin(n theta)
r=a cos (n theta)
(n=how many petals {if n isodd[#=n] if n is even [#=2n]}
Archimedes Spiral
r=a theta+b
Logarithmic Spiral
r=a^theta b
CONVERTING
when going from polar to rectangular you plug into
X=rcos(theta)
Y=rsin(theta)
and work out until you get a x point and a y point
when going from rectangular to polar you plug into
r=+/- squareroot X^2 +Y^2
and
Theta= (Y/X)
once youve solved for both of these you"ll plug into (+r, theta) (-r, theta)
i need help with a review of the formulas from the triangle section if anyone can help please do!
Monday, March 15, 2010
Devin's Reflection
Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
Devin's Reflection
CIRCLES
The equation of a circle in standard form is (x-h)^2-(y-h)^2=r^2 with the center being (h,k) and r being the radius.
Finding the intersection of a line and a circle:
1) solve linear equation for y
2) substitute in circle equation
3) solve for x
4) plug x in to get y value
(if x happens to be imaginary, there is no point of intersection)
EG:
(x-4)^2+(y+2)^2=16
center:(4,-2) radius:4
x^2+y^2+12y+16x-5=0
x^2+16x+64+y^2+12y+36=5+64+36
(x+8)^2+(y+6)^2=105
center:(-8,-6) radius:square root of 105
ELLIPSES
1) (x-h)^2/(length of x/2)^2 + (y-k)^2/(length of y/2)^2 =1
2)center is (h,k)
3) major axis has larger denominator
4) vertex is on major axis
5) focus is smaller denom squared = larger denom squared - focus squared
focus is on major axis
Graphing:
1) find center
2) major axis = plus or minus the square root of the bigger denom
3) vertex
4) other intercepts
5) focus
6) length of major axis = 2 square root of
7) length of minor axis = 2 square root of
8) graph
EG:
x^2/4+y^2/1=1
1) (0,0)
2) x
3) +/-2 (2,0) (-2,0)
4) +/-1 (0,1) (0,-1)
5) 1=4-c^2 c=+/-square root of 3 (sr3,0) (-sr3,0)
6) 2 square root of 4 = 4
7) 2 square root of 1
8) graph
HYPERBOLAS
1) (x+h)^2/(length/2)^2 - (y-k)^2/(length/2)^2 =1
OR
-(x-h)^2/(length/2)^2 + (y-k)^2/(length/2)^2 =1
2) center (h,k)
3) major axis is non-negative
4) vertex is the square root of non-negative denom
5) asymptotes y=+/-(square root of y)/(square root of x)x
6) focus^2 = x denom + y denom
focus^2 = vertex^2 + other denom
to sketch:
1) shape
2) center
3) major
4) minor
5) other intercept - none for hyperbolas
6) focus
7) asymptotes y=+/-square root of y/square root of x
8) vertex
9) sketch
A) draw a box using the vertex and +/-sr of other denom
B) draw diagonal through box corners
C) sketch a parabola on each vertex
D) label focus and asymptotes
EG:
x/36-y/9=1
2) (0,0)
3) x
4) y
5) none
6) c^2=36+9 c^2=45 c=sr45 (sr45,0) (-sr45,0)
7) y=+/-square root of 5/square root of 6
8) +/-sr36 = +/-6 (6,0) (-6,0)
9) sketch
The equation of a circle in standard form is (x-h)^2-(y-h)^2=r^2 with the center being (h,k) and r being the radius.
Finding the intersection of a line and a circle:
1) solve linear equation for y
2) substitute in circle equation
3) solve for x
4) plug x in to get y value
(if x happens to be imaginary, there is no point of intersection)
EG:
(x-4)^2+(y+2)^2=16
center:(4,-2) radius:4
x^2+y^2+12y+16x-5=0
x^2+16x+64+y^2+12y+36=5+64+36
(x+8)^2+(y+6)^2=105
center:(-8,-6) radius:square root of 105
ELLIPSES
1) (x-h)^2/(length of x/2)^2 + (y-k)^2/(length of y/2)^2 =1
2)center is (h,k)
3) major axis has larger denominator
4) vertex is on major axis
5) focus is smaller denom squared = larger denom squared - focus squared
focus is on major axis
Graphing:
1) find center
2) major axis = plus or minus the square root of the bigger denom
3) vertex
4) other intercepts
5) focus
6) length of major axis = 2 square root of
7) length of minor axis = 2 square root of
8) graph
EG:
x^2/4+y^2/1=1
1) (0,0)
2) x
3) +/-2 (2,0) (-2,0)
4) +/-1 (0,1) (0,-1)
5) 1=4-c^2 c=+/-square root of 3 (sr3,0) (-sr3,0)
6) 2 square root of 4 = 4
7) 2 square root of 1
8) graph
HYPERBOLAS
1) (x+h)^2/(length/2)^2 - (y-k)^2/(length/2)^2 =1
OR
-(x-h)^2/(length/2)^2 + (y-k)^2/(length/2)^2 =1
2) center (h,k)
3) major axis is non-negative
4) vertex is the square root of non-negative denom
5) asymptotes y=+/-(square root of y)/(square root of x)x
6) focus^2 = x denom + y denom
focus^2 = vertex^2 + other denom
to sketch:
1) shape
2) center
3) major
4) minor
5) other intercept - none for hyperbolas
6) focus
7) asymptotes y=+/-square root of y/square root of x
8) vertex
9) sketch
A) draw a box using the vertex and +/-sr of other denom
B) draw diagonal through box corners
C) sketch a parabola on each vertex
D) label focus and asymptotes
EG:
x/36-y/9=1
2) (0,0)
3) x
4) y
5) none
6) c^2=36+9 c^2=45 c=sr45 (sr45,0) (-sr45,0)
7) y=+/-square root of 5/square root of 6
8) +/-sr36 = +/-6 (6,0) (-6,0)
9) sketch
Devin's Reflection
The standard form for a circle equation is (x-h)^2+(y-k)^@=r^2. The center is (h,k).
Examples: Find the center
a. (x-3)^2+(y+7)^2=19
center (3,-7)
b. x^2+y^2-6x+4y-12=0
x^2-6x+ (9)+y^2+4y+ (4)=12+9+4
(x-3)^2+(y+2)^2=25
center (3,-2)
The r stands for the radius of the circle. When the equation is not in standard equation you have to complete the square to put the equation in standard equation. You can determine the radius of a circle, by using the distance formula and if you are given the center and a point.
Example: Find radius
a.(x-3)^@+(y+7)^2=19
radius square root of (19)
To find the intersection of a line and a cirlce
1. solve the linear equation for y
2. substitute in circle equation
3. solve for x
4. plug x-value into get y-value
*If your x=value is imaginary, then there is no point of intersection.
With ellipses, the main thing to know is the steps. They are:
1. Find center
2. Find major axis - big denom.
3. Find vertex - square root of big denom.
4. Find other intercepts - square root of small denom
5. Find Focus
6. Find length of major axis
7. Find length of minor axis
8. Graph
Just like the ellipses, to sketch the hyperbolas you must follow the steps.
First find:
1. shape
2. center
3. major axis
4. minor axis
5. 0ther int.
6. vertex
7. focus
8. asymptotes
9. then sketch
To sketch a hyperbola,do the following
1. Draw a box using the vertex and square root of the other denom
2. Draw diagonals through box
3. Sketch a parabola on each vertex
4. Label focus and asymptotes
Examples: Find the center
a. (x-3)^2+(y+7)^2=19
center (3,-7)
b. x^2+y^2-6x+4y-12=0
x^2-6x+ (9)+y^2+4y+ (4)=12+9+4
(x-3)^2+(y+2)^2=25
center (3,-2)
The r stands for the radius of the circle. When the equation is not in standard equation you have to complete the square to put the equation in standard equation. You can determine the radius of a circle, by using the distance formula and if you are given the center and a point.
Example: Find radius
a.(x-3)^@+(y+7)^2=19
radius square root of (19)
To find the intersection of a line and a cirlce
1. solve the linear equation for y
2. substitute in circle equation
3. solve for x
4. plug x-value into get y-value
*If your x=value is imaginary, then there is no point of intersection.
With ellipses, the main thing to know is the steps. They are:
1. Find center
2. Find major axis - big denom.
3. Find vertex - square root of big denom.
4. Find other intercepts - square root of small denom
5. Find Focus
6. Find length of major axis
7. Find length of minor axis
8. Graph
Just like the ellipses, to sketch the hyperbolas you must follow the steps.
First find:
1. shape
2. center
3. major axis
4. minor axis
5. 0ther int.
6. vertex
7. focus
8. asymptotes
9. then sketch
To sketch a hyperbola,do the following
1. Draw a box using the vertex and square root of the other denom
2. Draw diagonals through box
3. Sketch a parabola on each vertex
4. Label focus and asymptotes
Devin's Reflection
Parabolas:
how to find the axis of symmetry, vertex, focus, & directrx??
1.) to find the axis of symmetry: x = -b/2a
how to find the axis of symmetry, vertex, focus, & directrx??
1.) to find the axis of symmetry: x = -b/2a
2.) for the vertex: (-b/2a, f(-b/2a)) or use complete the square:
y = (x+a)^2 + b.....a & b are numbers and (-a,b) = vertex
3.) to find the focus: 1/4p= the coefficient of x^2 and then add p
Note:
*If opens up, add to y value from vertex, if opens down, subtract
*If opens right, add to x value to vertex, if opens left, subtract)
4.) directrix: is p units behind the vertex
Note:
*If opens up, subtract; if opens down, add from y-value of vertex.
*If opens right, subtract x-value
*If opens left, add x-value
Example: x^2 + 1
~vertex:
x = -b/2a
x = 0/2(1) = 0
0^2 + 1 = 1
(0,1)
~Focus:
1/4p = 1
4p = 1
p = 1/4
(0, 1 + 1/4)
(0, 5/4)
~directrix:
y = 1 - 1/4
y = 3/4
Devin's Reflection
TRIGONOMETRY
Angles
- measured in degrees
- to find minutes, multiply what is behind the decimal by 60
- to find seconds, multiply what is behind the decimal by 0 and divide by 300 to get decimal
- angles are measured in degrees and radian
degrees = radians times 180 over pi
- to find coterminal angles, add or subtract 360 degrees or 2 pi
- must use degrees symbol if in degree or its wrong
EG: 12.3 degrees... write as t
12 degrees .3 times 60
12 degrees 1.8 minutes
15.36 degrees
15 degrees .36 times 60
15 degrees 21 minutes .6 times 60
15 degrees 21 minutes 36 seconds
25 degrees 20 minutes 6 seconds
25 plus 20 divided by 60 plus 6 divided by 3600
25.335 degrees
15 degrees 26 minutes 15 seconds
15 plus 26 divided by 60 plus 15 divided by 3600
15.4375 degrees
36 degrees
36 divided by 180 pi
1/5 pi
pi/5
3 pi/4 times 180/pi
135 degrees
225 degrees
225 divided by 180 pi
5/4 pi
pi/9 times 80 divided by pi
20 degrees
2.4 times 180 divided by pi
137 degrees 30 minutes 35.535 seconds
245 degrees 15 minutes 300 seconds
2445 plus 15 divided by 60 plus 300 divided by 3600
245.3 degrees
Devin's Reflection
1)s=rθ
2)k=1/2r^2θ (k=1/2rs)
r = radius, θ = angle, s = arc length, k = area of sector
EG: A sector of a circle has an arc length of 6cm and an area of 75cm^2. Find its radius and measure of its central angle.
s=6cm
k=75cm^2
r=?
θ=?
k=1/2rs
75=1/2r6
75=3r
r=25cm
s=r
6=25θ
θ=6/25
sinθ=y/r
cosθ=x/r
tanθ=y/x
cscθ=r/x
secθ=x/y
cotθ=x/y
r=√(x^2+y^2)
EG: sin180° = y/r
at 180°, y=0
0/1 = 0
thus sin180°=0
EG: cosπ/2 = x/r
π/2 is at 90°
at 90°, x=0
thus cosπ/2=0
Trig Chart:
0°
sin0=0
cos0=1
tan0=0
csc0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reference Angles (must be between 0° and 90°)
1)find which quadrant angle is in
2)determine the sign in that quadrant (+ve or -ve)
3)subtract 180° until the angle is between 0° and 90° (0 and π/2)
1)find the reference angle using chart or calculator
2)find what quadrant you need to be in based on the sign of the value
3)use notes to move to that quadrant
To Move:
I to IV = make negative and add 360°
I to III = add 180°
I to II = make negative and add 180°
II to IV = add 180°
EG: sin^-1(-√2/2) = 45°
45 + 180 = 225° = θ
I to IV
-45 + 360 = 315°
θ = 225°, 315°
2)k=1/2r^2θ (k=1/2rs)
r = radius, θ = angle, s = arc length, k = area of sector
EG: A sector of a circle has an arc length of 6cm and an area of 75cm^2. Find its radius and measure of its central angle.
s=6cm
k=75cm^2
r=?
θ=?
k=1/2rs
75=1/2r6
75=3r
r=25cm
s=r
6=25θ
θ=6/25
sinθ=y/r
cosθ=x/r
tanθ=y/x
cscθ=r/x
secθ=x/y
cotθ=x/y
r=√(x^2+y^2)
EG: sin180° = y/r
at 180°, y=0
0/1 = 0
thus sin180°=0
EG: cosπ/2 = x/r
π/2 is at 90°
at 90°, x=0
thus cosπ/2=0
Trig Chart:
0°
sin0=0
cos0=1
tan0=0
csc0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reference Angles (must be between 0° and 90°)
1)find which quadrant angle is in
2)determine the sign in that quadrant (+ve or -ve)
3)subtract 180° until the angle is between 0° and 90° (0 and π/2)
1)find the reference angle using chart or calculator
2)find what quadrant you need to be in based on the sign of the value
3)use notes to move to that quadrant
To Move:
I to IV = make negative and add 360°
I to III = add 180°
I to II = make negative and add 180°
II to IV = add 180°
EG: sin^-1(-√2/2) = 45°
45 + 180 = 225° = θ
I to IV
-45 + 360 = 315°
θ = 225°, 315°
Devin's Reflection
The sin and csc both deal with the y factor and radius. The cos and sec both deal with the x factor and radius. The tan and cot both deal with the x factor along and the y factor. Radius is the square root of x(2) + y(2).
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
The trig functions are:
sin 0= y/r
cos 0= x/r
tan 0= y/x
csc 0= r/y
sec 0= r/x
cot 0= x/y
Unit Circle
2-(0,1)-90-pi/2 1-(1,0)-0,360-0,2pi
3-(-1,0)-180-pi 4-(0,-1)-270-3pi/2
The Trig Chart
0 sin 0=0 cos 0-1 tan 0=0
30 sin pi/6=1/2 cos pi/6= (3)/2 tan pi/6=(3)/3
45 sin pi/4=(2)/2 cos pi/4=(2)/2 tan pi/4=1
60 sin pi/3=(3)/2 cos pi/3= 1/2 tan pi/3=(3)
90 sin pi/2=1 cos pi/2=0 tan pi/2= underined
0 csc 0= undefined sec 0=1 cot 0= undefined
30 sin pi/6=2 sec pi/6=(2) cot pi/6=(3)
45 sin pi/4=(2) sec pi/4=(2) cot pi/4=1
60 sin pi/3=2(3)/3 sec pi/3=2 cot pi/3=(3)/3
90 sin pi/2=1 sec pi/2=undefined cot pi/2=0
Reference angles must be between 0 and 90, 0 and pi/2
1. Find which quadrant angle is in
2. Determine the sign in that quadrant (+ve or -ve)
3. Subtract 180 until the angle is between 0 and 90, 0 and pi/2
We also learned inverses.
To solve for an angle
1. simplify any expression
2. get the trig function by itself
3. take the trig inverse of both sides
4. use chart unit circle or calculator to find 1 angle
5. set up and find other quadrants with the same sign as the value.
Ex. sin -1((2)/2)=45 quadrants 1 and 2
To make from quadrant to quadrant:
1-3 make it -ve and add 360
1-2 make it -ve and add 180
1-4 add 360
2-4 add 180
Devin's Reflection
Rational Root therom
Example: f(x)= 2x^3 + 3x^2 - 8 + 3
Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2
*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2
Step 2: now you can plug all of the possible roots in your calculator to find the roots that work
you should get: (x - 1) (2x^2 + 5x + 3)
Example: f(x)= 2x^3 + 3x^2 - 8 + 3
Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2
*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2
Step 2: now you can plug all of the possible roots in your calculator to find the roots that work
- the zero will be: 1, 1/2, -3
you should get: (x - 1) (2x^2 + 5x + 3)
Step 4: slove further
(this can be factored...)
= (x - 1) (2x^2 + 5x + 3)
= (x - 1) (2x - 1) (x + 3)
(set x = 0 )
x = 1, 1/2, -3
Devin's Reflection
Ex. 2x^4-x^2-3=0
1. 2g^2-g-3=0
2. g(2g-3) 1(2g-3)
3. (g+1)(2g-3)
4. g=-1, 2g-3=0
5. g=-1, g=3/2
Normally I would stop right here I wouldn't plug the g's back into the equation.
6. g=x^2
7. x^2= -1, x^2= 3/2
8. x= +-i, x= (square root of 6)/2
And it took me the longest to understand the rational root theorem.
I'm not gone do an equation yall know it already.
And o yeah, I don't know how to do the backwards synthetic division. And I don't know when I need to use it. So if anyone can help me with that I will appreciate it.
I think I understand how to do inequalities. Its pretty simpe.
1. When its greater than
= -#<_<#
2. When its lesser than
= _<-# or _>#
The Domin and Range thing wasn't to difficult. You just have to remember to put the brackets where they should be.
1. brackets around shift
2. brackets arouund square roots with the limited domain and range
So I will keep on trying. Next week. Enjoy the 3 day weekend.
And if anyone can help me with the synthetic division thing, get at me.
1. 2g^2-g-3=0
2. g(2g-3) 1(2g-3)
3. (g+1)(2g-3)
4. g=-1, 2g-3=0
5. g=-1, g=3/2
Normally I would stop right here I wouldn't plug the g's back into the equation.
6. g=x^2
7. x^2= -1, x^2= 3/2
8. x= +-i, x= (square root of 6)/2
And it took me the longest to understand the rational root theorem.
I'm not gone do an equation yall know it already.
And o yeah, I don't know how to do the backwards synthetic division. And I don't know when I need to use it. So if anyone can help me with that I will appreciate it.
I think I understand how to do inequalities. Its pretty simpe.
1. When its greater than
= -#<_<#
2. When its lesser than
= _<-# or _>#
The Domin and Range thing wasn't to difficult. You just have to remember to put the brackets where they should be.
1. brackets around shift
2. brackets arouund square roots with the limited domain and range
So I will keep on trying. Next week. Enjoy the 3 day weekend.
And if anyone can help me with the synthetic division thing, get at me.
Devin's Reflection
How to Find the Inverse of a Function:
- Replace f(x) with y
- Reverse the roles of x and y
- Solve for y in terms of x
- Replace y with f-1(x)
Example 1 - f(x) = 2x + 3
- write the function as an equation: y = 2x + 3
- solve for x: x = (y - 3)/2
- now write f-1(y) as follows .
f -1(y) = (y - 3)/2 or f -1(x) = (x - 3)/2 - Check:
- f(f -1(x))=2(f -1(x)) + 3
=2((x-3)/2)+3 =(x-3)+3 =x - f -1(f(x))=f -1(2x+3)
=((2x+3)-3)/2 =2x/2 =x
Eample 2 - f(x) = √x + 4
- (x)^2 = (√y + 4)^2
- x^2 = y + 4
- y = x^2 - 4
- f-1(x) = (x^2 - 4)
- f(f-1(x)) = f(x^2 - 4) = √(x^2 - 4) + 4 = x
- f-1(f(x)) = f-1(√x + 4) = (√x + 4)^2 - 4 = x + 4 - 4 = x
Devin's Reflection
Logarithm Properties:
- logb MN = logb M + logb N
- logb M/N = logb M - logb N
- logb M^K = K logb M
- logb b^k = k (this one i don't get..maybe i copied it wrong)
- b^logb^k = k
Here are some examples:
1. log 2 + log 3 + log 4 = log 24 (mulitply: 2 x 3 x 4)
2. log 8 + log 5 - log 4 = log 10 (mulitply: 8 x 5 then divide: 40/4)
3. 2 ln 6 - ln 3 = ln 12 (raise 6 to the 2nd power = 36 the divided by 3 = 12)
4. log M - 3 log N = log M/ N^3
5. ln 2 + ln 6 - 1/2 ln 9 = ln 12/3 = ln 4
6. Expand logb MN^2....logb M + 2 logb N
7. Condense log 45 - 2 log 3....log (45/9) = log 5
8. Rewrite in exponetial form: log36 6 = 1/2....36^1/2 = 6
9. Rewrite in logarithmic form: 2^2 = 4....log2 4 = 2
Changing Bases: (Done when you can't solve a log)
- Rewrite it as an exponential
- Take the log of both sides
- Move the variable to the front
- then solve
(use the same steps when solving for x as an exponent when you can't write them as the same base)
examples:1. log5 10 = x
5^x = 10
log 5^x = log 10
x log 5 = 1
x = 1/log 5
2. 2^x = 7
log 2^x = log 7
x log 2 = log 7
x = log 7/log 2
(remeber b-rob might use random symbol so don't panic)
Devin's Reflection
This week we have gone over logarithems and functions. Logs have many dfferent principles and methods to solving them. For converting I have come up with my own littl system.
For logs to express them or simplify them, alll you do is rotate them. To express them all you do is rotate the variables and intergers to the right. The last number will be the exponent.
Example:
x^2=5
2log(2) 5
We have also gone threw the process of expanding and condensing logarithmic equations. We were also introduced to other symbols in which these equations can contain.
Example Expanding:
log2(x^2 y c^3/x y^4)
2log(2)x + log(2)y + 3log(2)c - log(2x - 4log(2)y
Example Condensing:
4logx - log2 - 2logc + logy + 3log4 - 2log6
log(x^4 y 4^3/2 6^2)
For logs to express them or simplify them, alll you do is rotate them. To express them all you do is rotate the variables and intergers to the right. The last number will be the exponent.
Example:
x^2=5
2log(2) 5
We have also gone threw the process of expanding and condensing logarithmic equations. We were also introduced to other symbols in which these equations can contain.
Example Expanding:
log2(x^2 y c^3/x y^4)
2log(2)x + log(2)y + 3log(2)c - log(2x - 4log(2)y
Example Condensing:
4logx - log2 - 2logc + logy + 3log4 - 2log6
log(x^4 y 4^3/2 6^2)
Sunday, March 14, 2010
Reflection
So we have an Exam in this class on the book Flatland.......haha ok old stuff
finding the center of Conics and the radius.
Steps:
1. The equation of a circle in standard form: (x-h)^2+(y-k)^2=r^2
the center=(h,k) and r=radius
2. If not in standard form, you must complete the square to put in standard form
3. Given the center and a point, you can use the distance formula to find the radius
Ex.: (x-3)^2+(y+7)^2
Center:(3,-7)
Radius:√19
Ex. with completing the square:
x^2+y^2-6x+4y-12=0
x^2-6x+__+4^2+4y+__=12
x^2-6x+9+4^2+4y+4=12+9+4
(x-3)^2+(y+2)^2=25
Center:(3,-2)
Radius:25
Does anyone know exactly what the exam is on in the book?
finding the center of Conics and the radius.
Steps:
1. The equation of a circle in standard form: (x-h)^2+(y-k)^2=r^2
the center=(h,k) and r=radius
2. If not in standard form, you must complete the square to put in standard form
3. Given the center and a point, you can use the distance formula to find the radius
Ex.: (x-3)^2+(y+7)^2
Center:(3,-7)
Radius:√19
Ex. with completing the square:
x^2+y^2-6x+4y-12=0
x^2-6x+__+4^2+4y+__=12
x^2-6x+9+4^2+4y+4=12+9+4
(x-3)^2+(y+2)^2=25
Center:(3,-2)
Radius:25
Does anyone know exactly what the exam is on in the book?
Trigonometric Identites:
csc = 1/sin x
sec = 1/cos x
cot x = 1/tan x
sin (-x) = -sin x
cos (-x) = cos x
csc (-x) = -csc x
sec (-x) = sec x
tan (-x) = -tan x
cot (-x) = -cot x
sin^2 x+cos^2 x = 1
1+tan^2 x = sec^2 x
1+cot^2 x = csc^2 x
sin x = cos(90*-x)
tan x = cot (90*-x)
sec x = csc (90*-x)
cos x = sin (90*-x)
cot x = tan (90*-x)
csc x = sec (90*-x)
tan x = sinx/cosx
cot x = cosx/sinx
sec = 1/cos x
cot x = 1/tan x
sin (-x) = -sin x
cos (-x) = cos x
csc (-x) = -csc x
sec (-x) = sec x
tan (-x) = -tan x
cot (-x) = -cot x
sin^2 x+cos^2 x = 1
1+tan^2 x = sec^2 x
1+cot^2 x = csc^2 x
sin x = cos(90*-x)
tan x = cot (90*-x)
sec x = csc (90*-x)
cos x = sin (90*-x)
cot x = tan (90*-x)
csc x = sec (90*-x)
tan x = sinx/cosx
cot x = cosx/sinx
Stephanie's Reflection
Limacon
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Saturday, March 13, 2010
Amy's Reflection #30
Idenities and Equations...
Steps for solving:
1. check Idenities
2.Use Algebra: combining, factoring, multiplacation, fraction, and sandwiching.
3. then go back to idenities
Example:
Simplify SecX-SinX TanX
use and identity for secX and tanX
1/cosX-sinX(sinX/cosX)
distribute sinx into ( sinx/cosx)
1-sin^2X/cosX
use the iden. and solve for cos^2X with 1-sin^2X
Cos^2 X/ CosX= Cosx
And here are some things you're gonna need to know for the test...
Reciprocal Relationships:
csc (theta)=1/sin (theta)
sec (theta) =1/cos (theta)
cot (theta) =1/tan (theta)
Relationships with Negatives:
sin (-theta) = -sin (theta) and cos (-theta) = -cos (theta)
csc (-theta) = -csc (theta) and sec (-theta)= -sec (theta)
tan (-theta)= -tan (theta) and cot (-theta)= -cot (theta)
Pythagorean Relationship:
sin² (theta) + cos² (theta) =1
1+tan² (theta) = sec² (theta)
1+cot² (theta) = csc² (theta)
Cofunction Relationships
sin (theta) = cos(90°- theta) and cos (theta) = sin(90°-theta)
tan (theta) = cot(90°-theta) and cot (theta) =tan(90°-theta)
sec (theta) = csc(90°-theta) and csc (theta)=sec(90°-theta)
Cautions:
**you can't divide trig functions to cancel them
Example:
2 cos (theta) / cos (theta) = cos² (theta)/ cos (theta)
**but you can move everything to one side & factor out a trig function
**or divide by a trig function to create a new one:
Example:
sinxtanx = 3sinx
sinxtanx - 3sinx = 0
sinx (tanx - 3) = 0
sinx = 0
x = sin^-1 (0)
x = 0, pie, 2pie
tanx -3 = 0
tanx = 3
x = tan^-1 (3)
x= 71.565 pie/180, 257.565 pie/180
Steps for solving:
1. check Idenities
2.Use Algebra: combining, factoring, multiplacation, fraction, and sandwiching.
3. then go back to idenities
Example:
Simplify SecX-SinX TanX
use and identity for secX and tanX
1/cosX-sinX(sinX/cosX)
distribute sinx into ( sinx/cosx)
1-sin^2X/cosX
use the iden. and solve for cos^2X with 1-sin^2X
Cos^2 X/ CosX= Cosx
And here are some things you're gonna need to know for the test...
Reciprocal Relationships:
csc (theta)=1/sin (theta)
sec (theta) =1/cos (theta)
cot (theta) =1/tan (theta)
Relationships with Negatives:
sin (-theta) = -sin (theta) and cos (-theta) = -cos (theta)
csc (-theta) = -csc (theta) and sec (-theta)= -sec (theta)
tan (-theta)= -tan (theta) and cot (-theta)= -cot (theta)
Pythagorean Relationship:
sin² (theta) + cos² (theta) =1
1+tan² (theta) = sec² (theta)
1+cot² (theta) = csc² (theta)
Cofunction Relationships
sin (theta) = cos(90°- theta) and cos (theta) = sin(90°-theta)
tan (theta) = cot(90°-theta) and cot (theta) =tan(90°-theta)
sec (theta) = csc(90°-theta) and csc (theta)=sec(90°-theta)
Cautions:
**you can't divide trig functions to cancel them
Example:
2 cos (theta) / cos (theta) = cos² (theta)/ cos (theta)
**but you can move everything to one side & factor out a trig function
**or divide by a trig function to create a new one:
Example:
sinxtanx = 3sinx
sinxtanx - 3sinx = 0
sinx (tanx - 3) = 0
sinx = 0
x = sin^-1 (0)
x = 0, pie, 2pie
tanx -3 = 0
tanx = 3
x = tan^-1 (3)
x= 71.565 pie/180, 257.565 pie/180
Tuesday, March 9, 2010
Stephen's Reflection
Ok so this week has been difficult cause of the tests mainly cause i dont remember a thing but there is stuff i do remeber from a more recent chapeter. Im going to give the identities formulas:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
tan -Θ= -tanΘ and cot -Θ= -cotΘ
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
I kinda forget alot of stuff in the last chapeter we did soo i need help with formulas.
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
tan -Θ= -tanΘ and cot -Θ= -cotΘ
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
I kinda forget alot of stuff in the last chapeter we did soo i need help with formulas.
Monday, March 8, 2010
Taylor Reflection 7 March
Due to the chapter tests we've been taking
I'll post some older notes applicable to the first couple chapter tests
Easy steps to work and sketch a polynomial function
1. Every X must be factored.
2. Put each X on the number line
3. Take each “ before and after”
4. * put in F(X)
5. Look for pos and neg
6. Sketch the graph
Now for the calculator
Press y = & enter the equation
Press graph Need max/min?
1. Press 2nd then trace
2. Pick max or min
3. Follow the “bound direction”
4. Press enter
5. Repeat steps 3 and 4
6. Then guess where the middle of the swoop is
7. Press enter
8. Put the x and y results into ()
Now repeat steps one through eight to solve for the other
And once you have the results placed in () you are done!
hope this helped.
what i could use help on is a review of finding intrigral co efficients when given two components
I'll post some older notes applicable to the first couple chapter tests
Easy steps to work and sketch a polynomial function
1. Every X must be factored.
2. Put each X on the number line
3. Take each “ before and after”
4. * put in F(X)
5. Look for pos and neg
6. Sketch the graph
Now for the calculator
Press y = & enter the equation
Press graph Need max/min?
1. Press 2nd then trace
2. Pick max or min
3. Follow the “bound direction”
4. Press enter
5. Repeat steps 3 and 4
6. Then guess where the middle of the swoop is
7. Press enter
8. Put the x and y results into ()
Now repeat steps one through eight to solve for the other
And once you have the results placed in () you are done!
hope this helped.
what i could use help on is a review of finding intrigral co efficients when given two components
Sunday, March 7, 2010
Reflection
WOW DUDE! I forgot about this test everyday thing and REALLY was not ready for that. I did pretty good on the first test most likely because I reallyyyyy learned that after I failed that test in the beginning of the year. So I always knew Logs pretty good:
If you have Log(5)25=2 you switch everything around to make sense. The base (5) goes first.
The answer should be: 5^2=25
If a log has no base the base is understood to be 10. If there is an e in the log then it is a natural log.
Something that I'm not too fresh with is completing the square. I do not know how but I just forgot how to do it for the test this week.
If you have Log(5)25=2 you switch everything around to make sense. The base (5) goes first.
The answer should be: 5^2=25
If a log has no base the base is understood to be 10. If there is an e in the log then it is a natural log.
Something that I'm not too fresh with is completing the square. I do not know how but I just forgot how to do it for the test this week.
Alicia's Reflection #29
Okay so all we have been doing since we watched the movie for Flatland is take our Chapter tests all the way back to chapter 1. Im just going to review some sequences and series for the upcoming chapter tests.
arithmetic sequences have addition and subtraction:
tn=t1+(n-1)d
> find the 28th term.
t28=3+(27)(2)
t28=3+54
t28=57
a geometric sequence is a sequence that has multiplication and division:
tn=t1*r^(n-1)
>find the 10th term
t10= 2*2^9
t10= 2*512
t10= 1024
Example: Find the formula for the nth term of the arithmetic sequence. 3,5,7
tn=3+(n-1)(2)
tn=3+2n-2
tn=1+2n
Example: Find the formula for the nth term of the sequence. 3,4.5,6.75
r=4.5/3=3/2
6.75/4.5=3/2
tn=3x(3/2)^n-1
arithmetic sequences have addition and subtraction:
tn=t1+(n-1)d
t28=3+(27)(2)
t28=3+54
t28=57
a geometric sequence is a sequence that has multiplication and division:
tn=t1*r^(n-1)
t10= 2*2^9
t10= 2*512
t10= 1024
Example: Find the formula for the nth term of the arithmetic sequence. 3,5,7
tn=3+(n-1)(2)
tn=3+2n-2
tn=1+2n
Example: Find the formula for the nth term of the sequence. 3,4.5,6.75
r=4.5/3=3/2
6.75/4.5=3/2
tn=3x(3/2)^n-1
alaina's makeup blog for 14 Feb 2010
Chapter 11 is pretty easy as long as you can recognize the shapes and equations. Also you have to know how to convert from polar to rectangular and from rectangular to polar as well as memorize the equations from sections 3 and 4.
Here are the names and equations to know:
**Limacon
r=a+b sin(θ) r=a+b cos(θ)
**Cardioid
r= a+ or -bsin (θ) r= a+ or - bcos (θ)
**Rose Curve
r=a sin(n θ) r=a cos (n θ)
n=how many petals there are
**Archimedes Spiral
r=a (θ)+b
**Logarithmis Spiral
r=ab^(θ)
**Common circle with center point at the pole
r=a sin (θ) r=a cos(θ)
When converting to recangular, use....
x=r cos (θ) or y=r sin (θ)
When converting to polar, use...
r= + or - √x2 + y2
take the tan inverse....
theta= tan-1(y/x)
Here are the names and equations to know:
**Limacon
r=a+b sin(θ) r=a+b cos(θ)
**Cardioid
r= a+ or -bsin (θ) r= a+ or - bcos (θ)
**Rose Curve
r=a sin(n θ) r=a cos (n θ)
n=how many petals there are
**Archimedes Spiral
r=a (θ)+b
**Logarithmis Spiral
r=ab^(θ)
**Common circle with center point at the pole
r=a sin (θ) r=a cos(θ)
When converting to recangular, use....
x=r cos (θ) or y=r sin (θ)
When converting to polar, use...
r= + or - √x2 + y2
take the tan inverse....
theta= tan-1(y/x)
alaina's makeup blog for 22 Feb 2010
I know all the formulast we have had to memorize for the most part.
They are:
Sine and Cosine Sum/Difference Formulas:
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
Basics:
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
and for the most part I can work the majority of the problems, I just get myself a little lost every not and again. And I have problems with the ones that call for the use of chapter eight.
I get a little lost in translation of the formulas with the new formulas
so if anyone knows any tricks for these that would be great.
They are:
Sine and Cosine Sum/Difference Formulas:
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
Basics:
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
and for the most part I can work the majority of the problems, I just get myself a little lost every not and again. And I have problems with the ones that call for the use of chapter eight.
I get a little lost in translation of the formulas with the new formulas
so if anyone knows any tricks for these that would be great.
alaina's makeup blog for 28 Feb 2010
Sequences
A sequence is simply a list of numbers.
There are two main types of sequences:
Arithmetic - where you add or subtract
Geometric - where you multiply
(*Note: division is considered Geometric. For example: If a sequence divides by three, it is considered to be multiplied by one-third.
Formulas to find a term:
Arithmetic
tn-t'+(n-1)d
n=term #
t'=first term
d=what you add
tn=term#_in sequence
Geometric
tn=t'∙r^(n-1)
r= what you multiply by
Recursive Definitions
A recursive Definition is a formula for a sequence that involves a previous term. [a(n-1)]
an= (an-1/3)
I'm having trouble with sigma notation. if anyone can help???
A sequence is simply a list of numbers.
There are two main types of sequences:
Arithmetic - where you add or subtract
Geometric - where you multiply
(*Note: division is considered Geometric. For example: If a sequence divides by three, it is considered to be multiplied by one-third.
Formulas to find a term:
Arithmetic
tn-t'+(n-1)d
n=term #
t'=first term
d=what you add
tn=term#_in sequence
Geometric
tn=t'∙r^(n-1)
r= what you multiply by
Recursive Definitions
A recursive Definition is a formula for a sequence that involves a previous term. [a(n-1)]
an= (an-1/3)
I'm having trouble with sigma notation. if anyone can help???
Alaina's blog, 6 March 2010
CHAPTER 10 REVIEW
Here are all the formulas to memorize from each section:
10-1
**Sum and difference formulas for cosine and sine:
cos (alpha + or - beta) = cos(alpha)cos(beta) - or + sin(alpha)sin(beta)
sin (alpha + or - beta) = sin(alpha)cos(beta) + or - cos(alpha)sin(beta)
**Use these formuals to rewrite a sum or difference as a product:
sin(x) + sin(y) = 2sin(x+y/2)cos(x-y/2)
sin(x) - sin(y) = 2cos(x+y/2)sin(x-y/2)
cos(x) + cos(y) = 2cos(x+y/2)cos(x-y/2)
cos(x) - cos(y) = -2sin(x+y/2)sin(x-y/2)
10-2
**Sum and difference formulas for tangent:
tan(α) + tan (β)= tan (α)tan (β)/1- tan (α) tan (β)
tan (α)- (β)=tan (α) tan (β)/1+ tan (α) tan (β)
10-3
**Double-angle and half-angle formulas
sin 2(α)== 2sin(α)cos(α)
cos 2(&alpha) = cos^2(α)-sin^2(α)=1-2sin^2(α)=2cos^2(α)-1
tan 2(α)== 2tan(α)/1-tan^2(α)
sin(α/2)= +- √(1-cos(α)/2) cos(α/2)= +- √(1+cos(α)/2)
tan(α/2)= +- √(1-cos(α)/1+cos(α))=sin(&alpha)/1=cos(α)/2
1-cos(α)/sin(α)
**Decimal- use half-angle formula to find alpha. multiply the decimal angle by 2.
I am having trouble knowing when to use what double or half angle formula. I have no clue which one to use because there are soo many similar ones. So if anyone can help me with that i would be happy! I also have trouble doing the problems when you have to make a triangle to find the angle for cos or sine. I dont understand what numbers go on the triangle, which quad the triangle goes in, and what side of the triangle the numbers go on.
Here are all the formulas to memorize from each section:
10-1
**Sum and difference formulas for cosine and sine:
cos (alpha + or - beta) = cos(alpha)cos(beta) - or + sin(alpha)sin(beta)
sin (alpha + or - beta) = sin(alpha)cos(beta) + or - cos(alpha)sin(beta)
**Use these formuals to rewrite a sum or difference as a product:
sin(x) + sin(y) = 2sin(x+y/2)cos(x-y/2)
sin(x) - sin(y) = 2cos(x+y/2)sin(x-y/2)
cos(x) + cos(y) = 2cos(x+y/2)cos(x-y/2)
cos(x) - cos(y) = -2sin(x+y/2)sin(x-y/2)
10-2
**Sum and difference formulas for tangent:
tan(α) + tan (β)= tan (α)tan (β)/1- tan (α) tan (β)
tan (α)- (β)=tan (α) tan (β)/1+ tan (α) tan (β)
10-3
**Double-angle and half-angle formulas
sin 2(α)== 2sin(α)cos(α)
cos 2(&alpha) = cos^2(α)-sin^2(α)=1-2sin^2(α)=2cos^2(α)-1
tan 2(α)== 2tan(α)/1-tan^2(α)
sin(α/2)= +- √(1-cos(α)/2) cos(α/2)= +- √(1+cos(α)/2)
tan(α/2)= +- √(1-cos(α)/1+cos(α))=sin(&alpha)/1=cos(α)/2
1-cos(α)/sin(α)
**Decimal- use half-angle formula to find alpha. multiply the decimal angle by 2.
I am having trouble knowing when to use what double or half angle formula. I have no clue which one to use because there are soo many similar ones. So if anyone can help me with that i would be happy! I also have trouble doing the problems when you have to make a triangle to find the angle for cos or sine. I dont understand what numbers go on the triangle, which quad the triangle goes in, and what side of the triangle the numbers go on.
Stephanie's Reflection
Sine and Cosine Sum/Difference Formulas:
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
Tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
cos(alpha+/-beta)=cos alpha cos beta-/+sin alpha sin beta )
sin(alpha+/- beta)=sin alpha cos beta +/-cos alpha sin beta
sin x+sin y=2sin(x+y/2)cos(x-y/2)
sin x-sin y=2cos(x+y/2)sin(x-y/2)
cos x+cos y= 2cos(x+y/2)cos(x-y/2)
cos x-cos y=-2sin(x+y/2)sin(x-y/2)
Tangent Sum/Difference Formulas:
tan(alpha+beta)=tan alpha+tan beta/1-tan alpha tan beta
tan alpha-beta=tan alpha-tan beta/1+tan alpha tan beta
Double-Angle/Half-Angle Formulas:
sin 2α=2sinα cosα
cos 2α=cos2α-sin2α=1-2sin2α=2cos2α-1
tan 2α=2tanα/1-tan2α
sin(α/2)=+/-√(1-cosα/2) cos(α/2)= +/-√(1+cosα/2)
tan(α/2)= +/-√(1-cosα/1+cosα)=sinα/1+cosα= 1-cosα/sinα)
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
Reciprocal Relationships:
cscΘ=1/sinΘ
secΘ=1/cosΘ
cotΘ=1/tanΘ
Relationships with Negatives:
sin -Θ= -sinΘ and cos -Θ= -cosΘ
csc -Θ= -cscΘ and sec -Θ= -secΘ
Tan -Θ= -tanΘ and cot -Θ= -cotΘ
Pythagorean Relationships:
sin²Θ+cos²Θ=1
1+tan²Θ=sec²Θ
1+cot²Θ=csc²Θ
Cofunction Relationships:
sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)
Saturday, March 6, 2010
Amy's Reflection #29
okay here's some stuff that might to be helpful on the upcoming test...
Completing the Square:
You can use completing the square to solve a quadratic equation when factoring doesn’t work. This method can only work when 1 is the coefficient of x².
For example:
x² + 6x - 2 = 0
* anytime you are solving a quadratic you’re finding x-intercepts
Move the constant term to the right side:
x² + 6x = 2
Take half of the coefficient on the x-term (divide it by two, and keeping the sign), and then square it. Add the squared value to both sides of the equation:
x² + 6x + 9 = -2 + 9
Convert the left-hand side to squared form. Simplify the right-hand side:
(x + 3)² = 7
* the # half of the coefficient goes in the parentheses.
Square-root both sides:
x + 3 = √7
Solve for "x =". Remember to put the "±" on the right side and that it gives you two solutions.
x = -3 ± √7
The two points for this solution are:
(-3 + √7) , (-3 -√7)
Rational Root therom
Example: f(x)= 2x^3 + 3x^2 - 8 + 3
Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2
*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2
Step 2: now you can plug all of the possible roots in your calculator to find the roots that work
the zero will be: 1, 1/2, -3
Step 3: use synthetic division to factor all of the roots that work
you should get: (x - 1) (2x^2 + 5x + 3)
Step 4: slove further
(this can be factored...)
= (x - 1) (2x^2 + 5x + 3)
= (x - 1) (2x - 1) (x + 3)
(set x = 0 )
x = 1, 1/2, -3
i hope that this helps..
Completing the Square:
You can use completing the square to solve a quadratic equation when factoring doesn’t work. This method can only work when 1 is the coefficient of x².
For example:
x² + 6x - 2 = 0
* anytime you are solving a quadratic you’re finding x-intercepts
Move the constant term to the right side:
x² + 6x = 2
Take half of the coefficient on the x-term (divide it by two, and keeping the sign), and then square it. Add the squared value to both sides of the equation:
x² + 6x + 9 = -2 + 9
Convert the left-hand side to squared form. Simplify the right-hand side:
(x + 3)² = 7
* the # half of the coefficient goes in the parentheses.
Square-root both sides:
x + 3 = √7
Solve for "x =". Remember to put the "±" on the right side and that it gives you two solutions.
x = -3 ± √7
The two points for this solution are:
(-3 + √7) , (-3 -√7)
Rational Root therom
Example: f(x)= 2x^3 + 3x^2 - 8 + 3
Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2
*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2
Step 2: now you can plug all of the possible roots in your calculator to find the roots that work
the zero will be: 1, 1/2, -3
Step 3: use synthetic division to factor all of the roots that work
you should get: (x - 1) (2x^2 + 5x + 3)
Step 4: slove further
(this can be factored...)
= (x - 1) (2x^2 + 5x + 3)
= (x - 1) (2x - 1) (x + 3)
(set x = 0 )
x = 1, 1/2, -3
i hope that this helps..
Monday, March 1, 2010
Taylor reflection for 28 february
Here are some more trig facts
Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
Taylor reflection for 28 february
Here are some more trig facts
Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi2
70 degs. = (0,-1) 3pi/2
360 degs. = (1,0) 2pi
sin=y/r
cos=x/r
tan=y/x
cot=x/y
sec=r/x
csc=r/y
SOHCAHTOA:
S = sin
O = opposite angle
H = hypotenuse
(sin = opposite/hypotenuse)
C = cos
A = adjacent angle
H = hypotenuse
(cos = adjacent/hypotenuse)
T = tan
O = opposite angle
A = adjacent angle
(tan = opposite/adjacent)
*the hypotenuse is opposite the right angle.
*A= 1/2 bh*
*To find the area of a non right triangle use this formula:
*A= 1/2 (leg)(leg)SIN(angle b/w)
*When you have a non right triangle that has pairs, use the law of sines:
Sin A/a = Sin B/b= Sin C/c
*All you are doing is setting up a proportion.
**Remember to solve for an angle, you have to take the inverse.
*To solve a triangle with no angles, use the Law of Cosines:
(opp leg)^2= (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg) Cos(angle b/w)
Stephanie's Reflection
Limacon
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
r = a+b sin theta
r = a+b cos theta
Cardioid
a-b sin theta
a-b cos theta
Rose
r = a sin n theta
r = a cos n theta
n is how many petals
Archimedes Spiral
r = a theta +b
Logarithmic Spiral
r=a^theta b
Converting
polar to rectangular
x=r cos theta
y=r sin theta
rectangular to polar
r=+/- sqrt x^2 + y^2
theta is (x/y)
Trig Chart:
0°
sin0=0
cos0=1
tan0=undefined
sec0=1
cot0=0
30°
sinπ/6=1/2
cosπ/6=√3/2
tanπ/6=√3/3
cscπ/6=2
secπ/6=2 √3/3
cotπ/6=√3
45°
sinπ/4=√2/2
cosπ/4=√2/2
tanπ/4=1
cscπ/4=√2
secπ/4=√2
cotπ/4=1
60°
sinπ/3=√3/2
cosπ/3=1/2
tanπ/3=√3
cscπ/3=2 √3/3
secπ/3=2
cotπ/3=√3/2
90°
sinπ/2=1
cosπ/2=0
tanπ/2=undefined
cscπ/2=1
secπ/2=undefined
cotπ/2=0
blog for TERRIO
I forgot to blog because of the weekend, and because Big Crunk didn't remind anyone this week. One thing I remember pretty good is converting degrees to radians and radians to degrees.
To find radians:
One thing I don't really remember how to do is graphing...
To find radians:
radians = degrees x pi/180
To find degrees:
degrees = radians x 180/pi
Ex:
360 degrees, convert to radians
360 x π/180 = 360π/ 180 = π/2Convert π/10 to degrees.
π/10 x 180/π = 18 degreesOne thing I don't really remember how to do is graphing...
Alicia's Reflection #27
Alrighty so we finally finished the book this week. I thought it was a good book but it was just very different. Im glad that i had a chance to read it though because i learned a little about the dimensions and it helped me to think outside the box on things. Okay so im going to review some trig to refresh myself on some things.
**The tangent formulas:
Sum Formula for Tangent:
tan(alpha + beta) = tan alpha + tan beta/1-tan alpha (tan beta)
Difference Formula for Tangent:
tan(alpha - beta) = tan alpha - tan beta/1+tan alpha (tan beta)
EXAMPLES:1)
Find the exact value of tan15 degrees+tan30 degrees/1-tan15 degrees
(tan30 degrees)
= tan(15 degrees + 30 degrees)
= tan(45 degrees)
= 1
Keep in mind that for these answers you have to know the trig chart!!
2)Find tan(alpha+beta)
tan alpha=2 tan beta=1
tan(alpha+beta) = tan alpha+tan beta/1-tan alpha(tan beta)
= 2+1/1-(2)(1)
= 3/-1
= -3
I look forward to watching the movie on flatland tomorrow so i can clear up some things that confuse me in the book. I could use some help with sigma notation!!
**The tangent formulas:
Sum Formula for Tangent:
tan(alpha + beta) = tan alpha + tan beta/1-tan alpha (tan beta)
Difference Formula for Tangent:
tan(alpha - beta) = tan alpha - tan beta/1+tan alpha (tan beta)
EXAMPLES:1)
Find the exact value of tan15 degrees+tan30 degrees/1-tan15 degrees
(tan30 degrees)
= tan(15 degrees + 30 degrees)
= tan(45 degrees)
= 1
Keep in mind that for these answers you have to know the trig chart!!
2)Find tan(alpha+beta)
tan alpha=2 tan beta=1
tan(alpha+beta) = tan alpha+tan beta/1-tan alpha(tan beta)
= 2+1/1-(2)(1)
= 3/-1
= -3
I look forward to watching the movie on flatland tomorrow so i can clear up some things that confuse me in the book. I could use some help with sigma notation!!
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