TRIG EXAM PT 1 TOMORROW!!
study study study
good luck everyone
for mw the main thing is to remember the formulas
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)
Half-Angle and Double-Angle Formulas:
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2
(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos
(alpha)=1-cos(alpha)/sin(alpha)
Recriprocal Relationships:
csc=1/ sin(theta)
sec=1/cos(theta)
cot=1/tan(theta)
Relationships With Negatives:
sin-theta= -sin(theta)
cos-theta= -cos(theta)
tan-theta= -tan(theta)
csc-theta= -csc(theta)
sec-theta= -sec(theta)
cot-theta= -cot(theta)
Pythogorean Relationships:
sin^2(theta)+cos^2(theta)=1
1+tan^2(theta)= sec^2(theta)
1+cot^2(theta)= csc^2(theta)
Cofunction Relationships:
sin(theta)= cos (90degrees-theta)
cos(theta)= sin (90degrees-theta)
tan(theta)= cot (90degrees-theta)
cot(theta)= tan (90degrees-theta)
sec(theta)= csc (90degrees-theta)
csc(theta)= sec (90degrees-theta)
Monday, May 3, 2010
Sunday, May 2, 2010
Alicia's Reflection #37
Okay so I've been studying for so long I have decided to stop and go to bed. Hopefully I remember all the formulas for chapters 8 and 10. The notecards that I bought from the math club in the beginning of the year were definitely helpful. Well, here is the trig chart. Tomorrow is the non calculator portion so if you do not know the trig chart you will probably fail :(.....
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
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
The 6 trig functions are also very helpful to memorize:
sin y/r csc r/y
cos x/r sec r/x
tan y/x cot x/y
Also to move from quadrants:
I-II make - then add 180
I-III just add 180
I-IV make - then add 360
Sin is positive in I and II... Negitive in III and IV
Cos is positive in I and IV... Negative in II and III
Tan is positive in I and III... Negative in II and IV
Goodluck to everyone tomorrow!!!
Terrio's Reflection
I'm done. I worked on these packets for like three straight weeks and studied a total of at least fifteen full hours this weekend for this huge Trig Exam and I'm still going to fail.
Area of Non-Right Triangle:
A=1/2(leg)(leg)sin(angle between)
Two sides of a triangle have lengths 7cm and 4 cm. The angle between the sides measures 73 degrees. Find the area of the triangle.
sides=7 and 4
angle between= 73 degrees
A=1/2(7)(4) sin(73)
A=13.388cm^2
Area of Non-Right Triangle:
A=1/2(leg)(leg)sin(angle between)
Two sides of a triangle have lengths 7cm and 4 cm. The angle between the sides measures 73 degrees. Find the area of the triangle.
sides=7 and 4
angle between= 73 degrees
A=1/2(7)(4) sin(73)
A=13.388cm^2
Stephanie's Reflection
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)
Half-Angle and Double-Angle Formulas:
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2
(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos
(alpha)=1-cos(alpha)/sin(alpha)
Recriprocal Relationships:
csc=1/ sin(theta)
sec=1/cos(theta)
cot=1/tan(theta)
Relationships With Negatives:
sin-theta= -sin(theta)
cos-theta= -cos(theta)
tan-theta= -tan(theta)
csc-theta= -csc(theta)
sec-theta= -sec(theta)
cot-theta= -cot(theta)
Pythogorean Relationships:
sin^2(theta)+cos^2(theta)=1
1+tan^2(theta)= sec^2(theta)
1+cot^2(theta)= csc^2(theta)
Cofunction Relationships:
sin(theta)= cos (90degrees-theta)
cos(theta)= sin (90degrees-theta)
tan(theta)= cot (90degrees-theta)
cot(theta)= tan (90degrees-theta)
sec(theta)= csc (90degrees-theta)
csc(theta)= sec (90degrees-theta)
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)
Half-Angle and Double-Angle Formulas:
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2
(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos
(alpha)=1-cos(alpha)/sin(alpha)
Recriprocal Relationships:
csc=1/ sin(theta)
sec=1/cos(theta)
cot=1/tan(theta)
Relationships With Negatives:
sin-theta= -sin(theta)
cos-theta= -cos(theta)
tan-theta= -tan(theta)
csc-theta= -csc(theta)
sec-theta= -sec(theta)
cot-theta= -cot(theta)
Pythogorean Relationships:
sin^2(theta)+cos^2(theta)=1
1+tan^2(theta)= sec^2(theta)
1+cot^2(theta)= csc^2(theta)
Cofunction Relationships:
sin(theta)= cos (90degrees-theta)
cos(theta)= sin (90degrees-theta)
tan(theta)= cot (90degrees-theta)
cot(theta)= tan (90degrees-theta)
sec(theta)= csc (90degrees-theta)
csc(theta)= sec (90degrees-theta)
Stephanie's Blog Topic Response
In order to tell what kind of graph you have, you look at the equation.
If the equation is r = a + b sin θ OR r = a + b cos θ, then you have a limacon.
If the equation is r = a - b sin θ OR r = a - b cos θ, then you have a cardiod.
If the equation is r = a θ + b, then you have an archimedes spiral whereas if the equation is r = a b ^ θ, then you have a logarithmic spiral.
If the equation is r = a sin(nθ) OR r = a cos(nθ), then it is a rose where n = the number of petals.
If the equation is r = a + b sin θ OR r = a + b cos θ, then you have a limacon.
If the equation is r = a - b sin θ OR r = a - b cos θ, then you have a cardiod.
If the equation is r = a θ + b, then you have an archimedes spiral whereas if the equation is r = a b ^ θ, then you have a logarithmic spiral.
If the equation is r = a sin(nθ) OR r = a cos(nθ), then it is a rose where n = the number of petals.
Stephen's Reflection
Ok so this week we have trig exam..one thing i forgot is shapes in graphs like circles and roses and spirals. I now know it so im going to give the formulas for each shape in a graph...
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
Ok so i forget stuff about logs like formulas and examples on how to work them so if i can get an example of each i would be happy
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
Ok so i forget stuff about logs like formulas and examples on how to work them so if i can get an example of each i would be happy
Saturday, May 1, 2010
Amy's Reflection #37
so we pretty much just reviewed for the test..here's some examples of the stuff we gonna have to know..
Example 1: Find the exact value of sin 80 cos 130+ cos 80 sin 130
Let alpha = 80 and beta = 130 then sin 80 cos 130 + cos 80sin 130 = sin alpha cos beta + cos alpha sin b
= sin (alpha + beta)
= sin (80+130)
= sin (210)
= sin (180 + 30)
= - sin 30
= -1/2
Example 2: Find the exact value of
cos^215 - sin^215 = cos(2alpha)
= cos(30)
= sqrt 3/2
Example 3: cos 15
alpha = 45, beta = 30
cos (alpha - beta) = cos 45 cos30 + sin45 sin 30
cos (45 - 30) = (√(2/2) (√(3/2) + (√(2/2) (1/2)
= √(√6 + √2)/all over 4
Example 4: (1 + cot^2) (1 - cos 2x)
(csc^2x) (1-cos2x)
(csc^2x) (1-(1 - 2sin^2x)
(1/sin^2x) (2sin^2x)
= 2
Example 4: Find the exact value of sin22.5
alpha=2(22.5)
alpha=45
**if you are given an angle with a decimal you use the half-angle formula. To find alpha, you multiply by two.
Example 5: Sum Formula for Cosine
cos 75 cos 15 + sin 75 sin 15
=cos (75-15) = cos 60 = 1/2
Example 6: Find the exact value of tan15 +tan30/1-tan15 (tan30)
= tan(15 + 30)
= tan(45)
= 1
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
i hope this helped..don't forget to study for our trig!!!
Example 1: Find the exact value of sin 80 cos 130+ cos 80 sin 130
Let alpha = 80 and beta = 130 then sin 80 cos 130 + cos 80sin 130 = sin alpha cos beta + cos alpha sin b
= sin (alpha + beta)
= sin (80+130)
= sin (210)
= sin (180 + 30)
= - sin 30
= -1/2
Example 2: Find the exact value of
cos^215 - sin^215 = cos(2alpha)
= cos(30)
= sqrt 3/2
Example 3: cos 15
alpha = 45, beta = 30
cos (alpha - beta) = cos 45 cos30 + sin45 sin 30
cos (45 - 30) = (√(2/2) (√(3/2) + (√(2/2) (1/2)
= √(√6 + √2)/all over 4
Example 4: (1 + cot^2) (1 - cos 2x)
(csc^2x) (1-cos2x)
(csc^2x) (1-(1 - 2sin^2x)
(1/sin^2x) (2sin^2x)
= 2
Example 4: Find the exact value of sin22.5
alpha=2(22.5)
alpha=45
**if you are given an angle with a decimal you use the half-angle formula. To find alpha, you multiply by two.
Example 5: Sum Formula for Cosine
cos 75 cos 15 + sin 75 sin 15
=cos (75-15) = cos 60 = 1/2
Example 6: Find the exact value of tan15 +tan30/1-tan15 (tan30)
= tan(15 + 30)
= tan(45)
= 1
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
i hope this helped..don't forget to study for our trig!!!
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