Taylor Reflection 31 JAN

This week we learned a very simple chapter
the chapter focused on recognizing patterns and using formulas to find a specific part of the pattern

the types of patterns are
Arithmetic
Geometric
Neither

Arithmetic - is addition
Geometric - is multiplication
(division is considered Geometric because if a sequence divides it is considered to be a fraction.)


Then come the 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

these formulas are used to find specific components of a pettern
its just a matter of memorizing what each variable of the formula represents and plugging in to solve.

What i dont understand comes along with the rest of what we learned
i think i am going to have a really hard time keeping track of which method to use on the test
if anyone has any tips for recognizing which lessons material to use that would be great

Dustin's Reflection

Last week we started chapter 13. This is the easiest chapter we've covered all year. Chapter 11 wasn't that bad either. I had some problems remembering how to convert from polar to rectangular and vice versa, but other than that it was easy. Here's and explanation of chapter 13.

There are two types of sequences:

Arithmetic (Addition or Subtraction)

Geometric (Multiplication)

There are formulas to find certain terms or how many terms are in the sequence.

Ex. The sequence is 1,3,5,7,9,...

Find the 25th term.

1. First off you find the pattern. In this case we add by 2 each time.
2. Now we find the info we need for the formula.

t1-first term in sequence

tn-the answer

n-numer in the sequence we want to find ( in this case 25)

d-what we add or subtract by

Next we plug into the formula which is: tn=t1+(n-1)d

tn=1+(25-1)2

tn=1+(24)2

tn=1+48

tn=49

Therefore 49 is the 25th term in the sequence.

Reflection #24

This week was AWESOME!!! Monday was my birthday and the stuff we learned all this week was very easy. I hope I still think that when it comes to taking the actual test. We learned about Series of Numbers or something like that. There's two different types, Arithmetic, and Geometric, but if it's neither one of these then it's Neither.

Here's the two formulas:

1. Arithmetic- tn=t1+(n-1)d

n=term # t1=first term d=what you add

2. Geometric- tn=t1*r^(n-1)

r= what you multiply by n=term # t1=first term d=what you add


Ex.
6, 12, 24,.....

this is a Geometric sequence so you would use the formula tn=t1*r^(n-1)

then you would have 6*2^(n-1)

if you wanted to know the 4th term you would plug the 4 into the n and this is what you would get.

6*2^(4-1)
6*2^(3)
6*8
48

The only trouble I think I might have is remembering the formulas in this Chapter.

Chapter 13 Sequences and Recursive Definitions

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)

Alicia's Reflection #24

Alrighty so this week we started chapter 13 which is our last chapter to cover before reading Flatland. It is a pretty simple chapter and should be easy to pass the test because its just sequences and series of numbers. Basically, just remember the formulas.

There are 2 main types of sequences:

1.) Arithmetic- tn*t1+(n-1)d

n=term # t1=first term d=what you add

2.) Geometric- tn=t1*r^(n-1)

r= what you multiply by

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

divide the second term by the first to get your r.

4.5/3= 3/2 r= 3/2

tn=3(3/2)^n-1

I could use some help on the problems that ask you specifically to find the 200th term for example. I could also use some help with the recursive definitions. THANKSS!!!

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

Amy's Reflection #24

ok this week we started chapter 13 and i have to say that it's fairly easy but we're all gonna make it complicated because that's what we do..ok here's an explaination and some examples..

13-1

1. sequence-list of numbers

2. two main types: 1). arithmetic-add or subtract 2).geometric-multiply

Formulas:

1. arithmetic-used to find a term: tn . t1 + (n-1)d

**n=term #, t1=first term, d=what you add, tn=term #

2. geometric: tn=t1 . r^(n-1)

**r=what you multiply by..

Examples:

1. 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 + 2

2. Find the formula for the nth term of the sequence: 3,4.5,6.75,..

**divide the 2nd term by the 1st term to find r

4.5/3 = 3/2 = r

tn = 3 . (3/2)^(n-1)

13-2

Formula for a sequence that involves the previous term: (an - 1)

Examples:

1. Find the recursive definition of: 81, 27, 9,3,...

an = an - 1/3

2. 1, 2, 6, 24, 120, 720, ....

n = 1: 1

n= 2: 2

n = 3: 6

an = n . an - 1

13 -3

Series-List of added or subtracted numbers

**Leave it as a list: do NOT add

Formulas:

1. Arithmetic: Sn = n(t1 + t2)/2

**Finds the sum of the first n terms

2. Geometric: Sn = t1 (1 -r^n)/1-r

Examples:

1. Find the sum of the first 25 terms of the series: 11 + 14 + 17 + 20 + ....

Sn = n (t1 + tn)/2

t25 = 11 + (24)(3)

Sn = 25 (11 + 83)/2

= 1175

2. Find the sum of the first 10 terms of the series: 2-6 + 18 - 54 +...

**This is a geometric sequence and that is because you have to add or subtract the same number for it to be an arithmetic sequence, got it??

r = -6/2 = -3

Sn = t1(1 - r^n)/1-r

= 2(1 -(-3)^10)/1 - (-3)

= 2(-59048)/2

= -29524

so there you go..that's pretty much what we went over this week..and now for what i dont get: i really dont understand how to do problems like #3 on the quiz..i know im being little vague but all i remember is that i had no idea know to do it...so if anyone remember the question (which i doubt anyone does) can ya help me with that??

Stephen's Reflection

Ok so this is a late blog soooo yea...It will be on chapter 13 on the formulas. There are 2 types of the formulas: arithmetic and geometric. The first one i will discuss is arithmetic.

The formula for arithmetic is tn=t1 + (n-1)d where n is the term number, t1 is the first term and d is what you add. For arithmetic, you will add or subtract numbers.

The other formula the the geometric formula. the formula for geometric is tn=t1-r^(n-1) where r is what you multiply by. For geometric formulas, you will multiply.

these are pretty easy formulas to remember but what i have problems with is finding the term numbers.

last week's blog

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.

alaina's makeup blog

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(theta) r=a+b cos(theta)

**Cardioid
r= a+ or -bsin (theta) r= a+ or - bcos (theta)

**Rose Curve
r=a sin(n theta) r=a cos (n theta)
n=how many petals there are

**Archimedes Spiral
r=a (theta)+b

**Logarithmis Spiral
r=ab^(theta)

**Common circle with center point at the pole
r=a sin (theta) r=a cos(theta)


When converting to recangular, use....

x=r cos (theta) or y=r sin (theta)

When converting to polar, use...

r= + or - √x² + y²

take the tan inverse....
theta= tan^-1(y/x)

Polar Functions

Limacon - looks like a 'lima' bean

 

r=a+b sin(Θ)

r=a+b cos(Θ)

 

Cardioid - looks like a heart - (cardio as in cardiovascular)

 

a-b sin(Θ)

a-b cos(Θ)

 

Rose

r=a sin(nΘ)

r=a cos (nΘ)

To find n:

For number of petals = p

If p is odd, n=p

If p is even n=2p

 

Archimedes Spiral - The thing that looks really trippy when you make it black and white, and no, it's not moving.

r=aΘ+b

 

Logarithmic Spiral – Looks like a nautilus

r=a^Θb

Taylor #twenty something blog

This week we mainly focused on two things and then we learned a little bit on another set of ideals that will be included in the test on tuesday.

the two major things we focused on were memorizing graph shapes and their formulas and converting between rectangular and polar and viseversa

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)



the only thing i am having trouble with is the converting problems where the answers are already given and you have to solve for the things that were plugged into the equation to get the given answer

Kane's Reflection

Yeah, this is my second blog. I pretty much get everything we did this week. My only problem is remembering formulas. I guess I need to study that stuff more. Anyway, here are the formulas:

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

Circle With Pole Center
r=c

Circle Intersecting Pole
r=a sin(theta)
r=a cos(theta)

I really just need to review over the notes and the chapter test. And memorize all the formulas of course.

Alicia's Reflection #23

Alrighty, so the SAINTS won!!! 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(theta) r=a+b cos(theta)

**Cardioid
r= a+ or -bsin (theta) r= a+ or - bcos (theta)

**Rose Curve
r=a sin(n theta) r=a cos (n theta)
n=how many petals there are

**Archimedes Spiral
r=a (theta)+b

**Logarithmis Spiral
r=ab^(theta)

**Common circle with center point at the pole
r=a sin (theta) r=a cos(theta)


When converting to recangular, use....

x=r cos (theta) or y=r sin (theta)

When converting to polar, use...

r= + or - squareroot of x^2 + y^2

take the tan inverse.... theta= tan-1(y/x)


CHAPTER TEST TUESDAY!!!! Goodluck :)

Amy's Reflection #23

we this week was all about chapter 11 so here are some of the stuff we went over & examples..

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..

Stephanie's Reflection

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

Circle With Pole Center
r=c

Circle Intersecting Pole
r=a sin(theta)
r=a cos(theta)

1. 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)

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

3. Double-Angle/Half-Angle Formulas
sin 2 alpha=2sin alpha cos alpha

• cos 2 alpha=cos^2 alpha-sin^2 alpha=1-2sin^2 alpha=2cos^2 alpha-1
• tan 2 alpha=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

if you have a decimal, us the half-angle formulas for alpha and then multiply the decimal angle by two

Reflection #23

Tomorrow is my birthday!!!!!! This week I think I messed up big time by not paying attention very well on Tuesday because I was not in the school mood. I struggled a good bit but by Thursday I pretty much had it. I think that the hardest thing is remembering all the formulas and knowing which one goes with what.

Heres the Formulas:

Limacon:r=a+b sin(theta) OR r=a+b cos(theta)
Cardioid:a-b sin(theta) OR r=a-b cos(theta)
Rose:r=a sin(n theta) OR r=a cos (n theta)
n=how many petals
Archimedes spiral:r=a theta+b
Logarithmic spiral:r=a b^theta
Circle with its center at the pole:r=c
Circle that intersects with the pole:r=a sin(theta) OR r=a cos(theta)

The thing I forgot how to do was go from Polar to Rectangular and Rectangular to Polar.

Stephen's Reflection

There are many different formulas in chapter 10 which include sum and difference formulas, double angle formulas and half angle formulas. The easiest of these three are the sum and difference formulas for cos, sin, and tan.

The sum and difference formula for sin is sin(a +/- b)=sin(a)cos(b)+/-cos(a)sin(b)

The formula for cos is cos(a +/- b)=cos(a)cos(b) -/+ sin(a)sin(b). The sign for cos will be opposite in teh parentheses which means if it is a - b then you will end up adding the cos and sin.

the sum and difference formula for tan is tan(a +/- b)=tan(a) +/- tan(b)/1 -/+ tan(a)tan(b)

The only thing i dont understand in this section is using the formulas with fractions with different denominators.

Taylor Reflection #22

I know all the formulast we have had to memorize

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 2 alpha=2sin alpha cos alpha
cos 2 alpha=cos^2 alpha-sin^2 alpha=1-2sin^2 alpha=2cos^2 alpha-1
tan 2 alpha=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\



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 equations
the two types of equations i am having trouble with are

A: the ones where is asks you to find something when it is _<_<_

and

B: the ones where it calls for the use of the 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

Alicia's Reflection #22

alrighty so we had a short week of school which was great!! We just reviewed for our big chapter 10 test that is tuesday so ill put some examples of problems we need to study.

Ex: Find the exact value of 2sin15cos15
=sin2a
=sin2(15)
=sin30

=1/2

Ex: Find the exact value of sin 15

alpha=45 beta=30

sin(45-30)=sin 45 cos 30- cos 45 sin 30
sin 15= (squareroot of 2/2) (squareroot of 3/2)-(squareroot of 2/2)(1/2)

sin 15= squareroot of 6 - squareroot of 2/4


Ex: Find the exact value of cos 75

alpha = 45 beta = 30
cos(a + b) = cos(a)cos(b)-sin(a)sin(b)
cos 75 = (squareroot of 2/2)(squareroot of 3/2) - (squareroot of 2/2)(1/2)

cos 75 = (squareroot of 6 - squareroot of 2)/4



okay so study all the formulas and it should be pretty easy if you know the formulas, trig chart, and unit circle. GOODLUCK everyone!!!

Flatland??

Can someone tell me what edition of the book we need?

Amy's Reflection #22

ok we had like only three days this week..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

alrighty then, i hope that's enough examples to help someone out..ok what i need help with is the problems you gotta make a triangle and some how suppose to figure it out..so thanks to whoever can help me out with that..

Stephanie's Reflection

1. 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)

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

3. Double-Angle/Half-Angle Formulas
sin 2 alpha=2sin alpha cos alpha

• cos 2 alpha=cos^2 alpha-sin^2 alpha=1-2sin^2 alpha=2cos^2 alpha-1
• tan 2 alpha=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

if you have a decimal, us the half-angle formulas for alpha and then multiply the decimal angle by two

Reflection #22

Sooo, this week in the SCC Vs. RA soccer game I went up for a header and some dudes teeth went in my head and split it open hahaha, so I missed the one thing I needed the most, the chapter 10 review. Oh well, I think I got this stuff for the most part. Right now I'm working on memorizing the hardest part in chapter 10 in my eyes.
10-3

Double-angle and half-angle formulas

sin 2(alpha) = 2sin(alpha)cos(alpha)

cos 2(alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1

tan 2(alpha) = 2tan(alpha)/1-tan^2(alpha)

sin(alpha/2)= +- √(1-cos(alpha)/2) cos(alpha/2)= +- √(1+cos(alpha)/2)

tan(alpha/2)= +- √(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos(alpha)= 1-cos(alpha)/sin(alpha)

For a Decimal:use the half-angle formula to find alpha then multiply the decimal angle by 2.

One thing I don't really understand, and it seems like a lot of people don't understand this is knowing when to use the double and half angle formula.

Stephen's Reflection

Ok so this chapter is a little confusing for me. I kinda understand the trig chart but i think thats why im confused or it can be because of working and solving the formulas. I dont really understand like how to solve formulas for all the things we have learned but i do know the formulas. The formulas i understand and know is the double and half angle formulas. There is db and half angle formulas for sin, cos, and tan.

Double angle formulas for sin2(a):
2sin(a)cos(a)
Half angle formulas for sin(a/2):
+/- square root of (1-cos(a)/2)

Double angle formulas for cos2a:
cos^2a-sin^2a, 1-2sin^2a, 2cos^2a-1
Half angle formulas for cosa/2:
+/- square root 1+cosa/2

Double angle formulas for tan2a:
2tana/1-tan^2a
Half angle formulas for tana/2:
+/- square root of 1-cosa/1+cosa, sina/1+cosa, 1-cosa/sina

Those are the formulas for db and half angles for sin, cos, and tan. the thing i dont understand is solving them in a problem.

Stephanie's Reflection

Chapter 10

1. 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)

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

3. Double-Angle/Half-Angle Formulas
sin 2 alpha=2sin alpha cos alpha

• cos 2 alpha=cos^2 alpha-sin^2 alpha=1-2sin^2 alpha=2cos^2 alpha-1
• tan 2 alpha=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

if you have a decimal, us the half-angle formulas for alpha and then multiply the decimal angle by two

Alicia's Reflection #21

Okay so we moved on to chapter 10 last week. Its really not that hard as long as you memorize all the formulas for each section. 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(alpha) + (beta)= tan (alpha) + tan (beta)/ 1-tan (alpha) tan (beta)

tan (alpha) - (beta)=tan (alpha) - tan (beta)/ 1+ tan (alpha) tan (beta)

10-3

**Double-angle and half-angle formulas

sin 2(alpha) = 2sin(alpha)cos(alpha)

cos 2(alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1

tan 2(alpha) = 2tan(alpha)/1-tan^2(alpha)

sin(alpha/2)= +- squareroot(1-cos(alpha)/2) cos(alpha/2)= +- sqrt(1+cos(alpha)/2)

tan(alpha/2)= +- squareroot(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos(alpha)=

1-cos(alpha)/sin(alpha)

**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.

Taylor Reflection #21

I was sick alot last week so i really am still grasping the lessons of the past week
the formulas i solve and use well are the

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)

and

Rewriting 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)


the one type of problem that i really need help with are the ones that say solve for an equation when equal to _<_<_

if someone could please please explain how to work problem like this and possibly give an example that would be so great.

Alaina's reflection

Okay, so seing as I missed a bunch, I got myself completely lost. So I somewhat understand what we learned this week. Umm.. I understand all of the first part of 10-1. I think it is the addition and subtraction properties. After that, I get lost, especially with 10-3. I understand that you are supposed to plug a degree in for alpha and convert it back to one of the origional equations, but I do not understand what to do when the degree plugged in for alpha is not on the unit circle or trig chart. I really do not understand much about this section so any help would be much obliged. (:

thanksss

Refection #21 I Think

NO SCHOOL TOMORROW!!!!!!!!!! It's about time we get a day off, Lutcher is already on like their third day of no school since the holidays ended. This week was really easy. I'm not gunna lie though, I think it's probably because I did my homework everyday this week for the first time this year. The formulas I think I picked up the best were the Tan. formulas:
Tan (α + β)=tan α + tan β/1-tan α tan β
Tan (α - β)=tan α - tan β/1+tan α tan β

Say you get tan 15degrees
You would use two angles from the trig chart that add or subtract to get 15 degrees (45 and 30). You plug them into the α & β parts of the problem then solve it. So here is what you would get.

Tan (45 - 30)=tan 45 - tan 30/1+tan 45 tan 30
=tan √2/2- tan√3/3 /1+tan √2/2 tan√3/3
=3(√3-3)/3(√3+3)
=(√3-3)/(√3+3)

The only thing I really dont understand from this week is the four random formulas we wrote down but never used.
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
What is this????

Amy's Reflection #21

hey, no school tomorrow (haahaa i got a four day weekend)!!!ok well anyway, this week we started and finished chapter 10...and god knows there is a lot of formulas to remember..so here they are:

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)

Rewriting 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)
**we didn't use these formulas for anything..so i got no idea where to use them..

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)

now here are some examples:

1. tan α = 2 and tan β=1, find tan (α - β)

= tan α + tan β/1-tan α tan β

=2+1/1-(2)(6)

=3/-1

=-3

2. Find the exact value of: tan 15+tan 30/1-tan 15 tan 30

tan α = 2 and tan β=1

find tan (α - β)

= tan (15 + 30)

=tan (45)

=1

3. Find the exact value of sin 15degrees

*exact value means you use your trig chart
*think of two numbers from the trig chart can either add or subtract to give you 15
*since it's (45-30), you would look for the formula that uses sin

sin (a-B) = sin a cos B - cos a sin B

* plug #s into equation..

a=45 degrees B=30 degrees

sin (45-30) = sin 45 cos 30 - cos 45 sin 30

sin (a-B) = sin a cos B - cos a sin B

sin (45-30) = sin 45 cos 30 - cos 45 sin 30

sin 15 = (square root of 2 over 2)(square root of 3 over 2) - (square root of 2 over 2)(1/2)

sin 15 degrees = (square root of 6 over 4) - (square root of 2 over 4)

= square root of 6 - square root of 2 all over 4

and peole remember we wont be allowed to use a calc. with trig functions so you might wanna pick up one that doesnt so ya add wrong on the upcoming test...ok i need help in is the last section..can someone explain 10-4 to me?

What is a Trigonometric Function?

So as you may have learned in class, trigonometric functions can be related to find the sides of a triangle, the parts of a circle, or a wave when the function is graphed. But if you are curious as I am, then you must be wondering how such a seemingly abstract concept such as sine (something that is supposedly a wave when graphed) can be related to the parts of a circle or even more puzzlingly, the sides of a triangle. I have included this animation in an attempt to explain. Note: if the animation doesn't play, click here.

Stephen's reflection

Ok so another thing i really understand is logs...and log properties. I guess ima try to explain that to the best of my ability because im really good at logs. so there are a few logs properties..

logb MN = logb M + logb N
logb M/N = logb M - logb N
logb M^K = K logb M
logb b^k = k
b^logb^k = k
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
Example:

log5 10 = x

5^x = 10

log 5^x = log 10

x log 5 = 1

x = 1/log 5

And there is still something i dont understand which is conics like formulas for circles and ellipses and stuff like that. sooo yea...

taylor holiday reflection #3

ill review more of the recent information for this reflection
but i must say the stuff we have seen is old news now
we know it
its simple for us to work with
but im nervous for whats to come with the last half of the school year

Degrees to Radians is: degrees*pi/180
Radians to Degree is: radians*180/pi

Unit Circle:
90 degs. = (0,1) pi/2
180 degs. = (-1,0) pi
270 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)

Devin's Reflection

The Law of Sines

Sin A/a = Sin B/b = Sin C/c

-the only way to use this formula, is unless you have a pair (meaning an angle and its corresponding side)
-you plug in the factors that can be accounted for
-you use cross-multiplication to get the formula in a horizontal equation
-you use algebra to finish the equation

The Law of Cosines

opposite leg^2 = adjacent leg^2 + other leg^2 -2(adjacent leg)(other leg)cos*

-this equation is a guide to help solving this certain types of triangles
-plug in the factors that you know of into the equation
-if you have to, you are allowed to renovate the equation to be accessible to various occasions
-you use algebra to finish the equation

Area of a triangles that are not Right angles

1/2(leg)(leg)sin*

SOHCAHTOA

Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
Csc = Hypotenuse/Opposite
Sec = Hypotenuse/Adjacent
Cot = Adjacent/Opposite

Stephanie's Reflection

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°

Unit Circle

sin theta = y/r

cos theta = x/r

tan theta = y/x

csc theta = r/x

sec theta = x/y

cot theta = x/y

r=sqrtx^2+y^2

90 pi/2
180 pi
270 3pi/2
360 2pi

sin pos in one and two and neg in three and four

cos pos in one and four and neg in two and three

tan pos in one and three and neg in two and four

cot pos in one and neg in two three and four

sec pos in one and neg in two three and four

csc pos in one and two and neg in three and four

AREA OF A NON-RIGHT TRIANGLE
A=1/2(leg)(leg)sin(angle between)

RIGHT TRIANGLES

• hypotenuse opposite angle
• a=1/2bh
• SOHCAHTOA

sin theta=opposite/hypotenuse
cos theta=adjacent/hypotenuse
tan theta=opposite/adjacent

LAW OF SINES
sinA/a sinB/b sinC/c

• used when you know pairs in a non-right triangle
• you are setting up proportions

Right Triangles

• A=1/2bh
• SOHCAHTOA
• sinΘ=opposite leg/hypotenuse
• cosΘ=adjacent leg/hypotenuse
• tanΘ=opposite leg/adjacent leg

Law of Cosines
(opposite leg)²=(adjacent leg)² + (other leg)² - 2(adjacent leg)(adjacent leg)cos°
EG: x, 5, 6 angle = 35°
x²=5²+6²-2(5)(6)cos35°
x=√(5²+6²-2(5)(6)cos35°)
x≈3.443

Area of Inscribed Shapes
A=nr²sinΘcosΘ

Terrio's New Years Blog

My New Years was CRAZY!!! First of all me and Remi decided to through the biggest party ever thrown in the empty lot next to my house. We ended up having about 300 people show up then the cops came. Three days later me and Remi went mud riding in his truck in back of old 51 and we sunk his truck so we spent the rest of our holidays trying to get his truck unstuck. Now his truck doesn't work and he has to get it fixed :\ One thing that just popped into my head is how to convert degrees to radians and radians to degrees.

Degrees & Radians:

Degrees to radians= Degree times pi/180

Radians to degrees= Radians times 180/pi

So if you have 360 degrees you would multiply 360 by pi/180 and it would be 360pi/180. Then you simplifie it and you would get 2pi/1

If you have 2pi you would multipy 2pi by 180 over pi and you would get 360pi/1pi. When you simplifie this it would equal 360 degrees.

So my Christmas holidays were pretty good. I went on the levee every night to have a fire with my friends, I had a fire during the day with Remi, and I helped people build bonfires on the levee. I still haven't gotten my Christmas present because my phone contract does not expire till January 6th. I can't lie though, I forgot how to do a good bit of things in math, but I do remember one thing. I remember how to solve for x in a log.

1. log5 25 = x
The 5 would be the number raised to an exponent. The x would be the exponent, and 10 would be what 5 raised to the exponent equals. So it should look like this:
5^x = 25
x would equal 2

Alicia's Reflection #20 (Christmas Holiday)

Okay so my holidays went by super fast. We go back to school tomorrow already :( I going to reflect on some stuff from chapter 2.

Polynomials and function notation

degree: the highest exponent

0- constant
1- linear
2-quadratic
3-cubic
4-quartic
5-quintic

** No variable can be in the denominator for a polynomial

**A root, zero, and x-intercept are the same thing

**Polynomial- equation with only addition and subtraction of terms

**Leading term- term with highest degree

**Function notation f( ) = plug in what is in ( ) for x.

**Synthetic division- used to factor.

Ex.) x^9+5x^7+4x^10+9x^2+4

A.) Is the equation a polynomial? yes

B.) What is the degree? 10

C.) What is the leading term? 4x^10

D.) What is the leading coefficient? 4


Ex.) f(x) = x^3+5x^2+5x-2

A.) find f (2)= (2)^3 +4(2)^2+1
= 8+4(4)+1
=8+16+1
= 25


Hope everyone had a good holidays....See yall tomorrow!

Stephen's reflection

Something i really understand is identities. This is one of the eastiest things for me because i know it. You simplify trig equations by using both identities and algebra. Here are some of the relationships you need to look for in order to solve the equations.

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)

When you get an equation you have to first check to see if there are any identities you can use, if not you go to algebra, after that you go back to your identities and finish the problem. This is really easy you just need to memorize the relationships.

I pretty much forgot alot of stuff like solving conics...yea....and solving problems when you have to find axis of symmetry and stuff like that with the lengthy equations so i need help with that.

Stephen's Reflection...last weeks make up

Ok so this vacation was pretty good and as you can see i pretty much forgot about blogs.... I forgot most of the math but it shouldnt be hard to catch up. Soo im goin to explain something i pretty much understand which is logs and log properties. There are five log properties:

1. logbMN=logbM+logbN
2. logbM/N=logbM-logbN
3. logbM^k=KlogbM
4. log(b)B^k=k
5. b^logbK=k

If a question tells you to expand logbMN^2, then the answer is logbM+2logbN. If a problem tells you to condense log45-2log3 then you make it log45/9 which = log5. Expand means to make it bigger and condense means to make it smaller.

There are a few things i dont understand. One is remembering conics and how to solve them so if anyone can explain that for me id be thankful. Two i have trouble remembering the simple stuff in chap 1 like the quad formula and how to complete the square.

aMY'S rEflection #20 (second holiday blog)

hey guys, we're suppose to post a total of 3 blogs over the holidays, right? well anyway, if you dont really understand logs by now here is lil something that will hopefully help you out...

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

(we all know ms Robinson likes to use some crazy symbols ..so don't panic)

oh btw, HAPPY NEW YEARS EVERYONE!!!

Taylor holiday reflection #2

graphing parabolas was from the very first chapter
refresher anyone?

**#1

you need to see if the parabola will open up or down. think of it this was: if the first thing you see in the equation is a negative sign relate that to which way negative numbers go on a graph or think "if some thing is negative you get a thumbs down" like wise "if something is positive it gets a thumbs up" so first thing you see at the front of the equation is a negative sign? thumbs down therefore the parabola opens down. If the first thing you see is a positive number? thumbs up therefore the parabola opens up.(using analogies like this is good for memory. If you start thinking in terms of analogies you get faster at retaining information)

**#2

deciding the number of X intercepts is also an easy remembering problem to fix. first you need to answer the problembsquared - 4(a)(c)as you said you are very good at plugging in this formula because you have remembered it well.look at your answer to that and remember: positive answer is two x interceptsnegative answer is nonezero for an answer is one X interceptits better to have two than none so POSITIVE thing to have TWONEGATIVE thing to have NONE(i dont have a trick to remember zero.. i think its just a process of elimination thing.. if i didnt get a positive answer or a negative anser that means its not two x intercepts nor is it no X intercepts,, well that means its one X intercept)

**#3

to find an x intercept you solve for Xit says that in your question"find X intercept"remember "find X"(dont forget to put answer into point form. when solving for x you will always wind up having to square root. you know this meas the answer will be +/-. be sure to show this when convertine to point form. {I.E. (#,0) & (-#,0)} in many of the problems we had there was also a matter of carring a number to the other side. this is no big deal you just tack it on also. for example... if you ended with X-2= +/- square root 6/2 you would add 2 to both sides and put in point form. therefore you'd have (squareroot 6/2+2,0) & (- squareroot 6/2+2,0))


**#4

y- intercept is just taking the 0 in the y spot for the last answer and plugging it into the x spot in the equation. which then leaves you only the Y variable to solve for. Remember: "Find Y intercept""find Y"common sense will tell you the only way to do that is to plug something into the X spot.. and i told you what to plug in

**#5

Axis of semmatry is a simple conversion formula you'll have to memorize the same way you did for the quadratic formula. by writing it down everytime you solve for axis of semmatry until you see the formula in your sleep.the Formula (in case it isnt written down) is X= -b/2(a) (the a and b plug ins of course come from the original equation)your answer will be the point to put your DOTTED LINE on. because this formula solves for X you know it will pass through that point on the X line. You also know its a vertical line. so no worries.

**#6

the vertex is also just a matter of plugging in remember this step follows the step ahead of it so it retains the answer for X that means half of your vertex is solvedyou already have your X point for the vertex answer that answer is also plugged into the original equation which again leaves you to solve for y.this means you now have your vertex point because you solved for X in step 5 and your answer after plugging that in gave you the Y

FINALLY!

now that you have turned everything into points its just a matter of locating them and marking them all on your graph.After each point is marked connect the dots.just as a quick check look back and see if your parabola is supposed to open up or down if your graph matches then congratulations! everything seems to have gone well.

i know everything i've given you is alot to remember. just write down the hints and keep the sheet as a reference. it doesnt have to be word for word. just putting "step one- opens up or down? work {b-4ac} *positive thumbs up *negative thumbs down"

Taylor holiday reflection #1

from now on for the exams we will have to know the trig chart
i figured id share my way of remembering it

TRIGCHART

0° *sin 0= 0 *cos 0= 1 *tan 0= 0 *csc 0= undefined *sec 0= 1*cot 0= undefined
30° * sin π/6= 1/2 *cos π/6= √3/2 *tan π/6= √3/3 *csc π/6= 2 *sec π /6= 2 √3/3*cot π/6= √3
45° *sin π/4= √2/2 *cos π/4= √2/2 *tan π/4= 1 *csc π/4= √2 *sec π/4= √2 *cot π/4= 1
60° *sin π/3= √3/2 *cos π/3= 1/2 *tan π/3= √3 *csc π/3= 2 √3/3 *sec π/3= 2 *cot π/3= √3/2
90° *sin π/2= 1 * cos π/2= 0 *tan π/2= undefined *csc π/2= 1 *sec π/2= undefined *cot π/2= 0

I also discovered an easier way to memorize the chart

First you would have to set up the chart differently than it was given

It would have to look like this
Sin Cos Tan Csc Sec Cot
30

45

60

90

Just like a graph and then you fill in the parts for each
If you can memorize the sin column for the chart you can figure out the rest of the chart

Cos is just sin backwards
Tan is just sin/cos
Csc is just 1/sin

Etc. You just have to find the relations of each