10-2

10-2 was super easy because there were only two formulas to remember and you only need tan for them. I was going to do 10-3 but I did that last week lol.

The formulas are:
tan(a+b) = tan(a)+tan(b)/1-tan(a)tan(b)

tan(a-b) = tan(a)-tan(b)/1+tan(a)tan(b)

The sign on top is the same as in what you are doing, and the bottom sign is opposite of the top.

Ex. 1

Suppose tan(a) = 1/2 and tan(b) = 3/4. Find tan(a+b) and tan (a-b)

tan(a+b) = tan(a)+tan(b)/1-tan(a)tan(b)

Replacing, you get (1/2)+(3/4)/1-(1/2)(3/4)

Simplifying, it becomes (10/8)/1-(3/8) = (10/8)/(5/8) = 80/40 = 2

tan(a-b) = tan(a)-tan(b)/1+tan(a)tan(b)

Replacing, you get (1/2)-(3/4)/1+(1/2)(3/4)

Simplifying, it becomes (-2/8)/1+(3/8) = (-2/8)/(11/8) = -16/88 = -2/11

Double and Half Angle Formulas 10-3

This week in Brobs class we did double and half angle formulas which is just the formulas from 10-3:
1. (theta)=2sin(theta)cos(theta)
2. cos2(theta)=cos^2(theta)-sin^2(theta)
3. cos2(theta)=1-2sin^2(theta)
4. cos2(theta)=2cos^2(theta)-1
5. tan2(theta)=2tan(theta)/1-tan^2(theta)
6. sintheta/2=+/-square root of 1-cos(theta)/2
7. costheta/2=+/-square root of 1+cos(theta)/2
8. tantheta/2=+/-square root of 1-cos(theta)/1+cos(theta)
9. tantheta/2=sin(theta)/1+cos(theta)
10. tantheta/2=1-cos(theta)/sin(theta)


Example 1:
Simplify the given expression.
2tan25dgrees/1-tan^2(25degrees)
=tan2(25)
=tan50degrees


Example 2:
Simplify the given expression.
square root of 1-cos80degrees/2
=sin(theta/2)
=sin(80/2)
=sin40degrees

Example 3:
angleA is an acute angle.
If sinA=5/13, find sin2A and cos2A
x^2+5^2=13^2
x^2+25=169
x^2=144
x=12
cosA=12/13 tanA=5/12
sin2A=2(5/13)(12/13)
sin2A=120/169
cos2A=1-2(5/13)^2
=1-2(25/169)
=1-50/169
cos2A=119/169

Example 4:
angleA is an acute angle.
If tanA=4/2, find cos2A and tan2A
2^2+1^2=r^2
4+1=r^2
5=r^2
r=square root of 5
sinA=square root of 5/5 cosA=2square root of 5/5
cos2A=1-2(square root of 5/5)^2
=1-2(5/25)=1-10/25
=15/25
cos2A=3/5
tan2A=2(1/2)/1-(1/2)^2=2/2=1/1-2/4
=1/1-1/2=1/1/2
tan2A=2

---Danielleee

Double-Angle & Half-Angle Formulas!

**These would be your Double and Half Angle Formulas:
sin2(theta)=2sin(theta)cos(theta)
cos2(theta)=cos^2(theta)-sin^2(theta)
cos2(theta)=1-2sin^2(theta)
cos2(theta)=2cos^2(theta)-1
tan2(theta)=2tan(theta)/1-tan^2(theta)
sintheta/2=+/-square root of 1-cos(theta)/2
costheta/2=+/-square root of 1+cos(theta)/2
tantheta/2=+/-square root of 1-cos(theta)/1+cos(theta)
tantheta/2=sin(theta)/1+cos(theta)
tantheta/2=1-cos(theta)/sin(theta)

Example 1:
Simplify the given expression.
2cos^210degrees-1
=cos2(10)=cos20degrees

Example 2:
Simplify the given expression.
4tanB/1-tan^2B
=2(2tanB/1-tan^2B)
=tan2(B)=2tan2B

Example 3:
Simplify the given expression.
2sin35degrees(cos35degrees)
=sin2(35degrees)
=sin70degrees

Example 4:
Simplify the given expression.
2tan25dgrees/1-tan^2(25degrees)
=tan2(25)
=tan50degrees

Example 5:
Simplify the given expression.
1-2sin^2(x/2)
=cos2(x/2)
=cosx

Example 6:
Simplify the given expression.
square root of 1-cos80degrees/2
=sin(theta/2)
=sin(80/2)
=sin40degrees

Example 7:
Simplify the given expression.
2cos(2pi/8)-1
=cos2(pi/8)
=cos2(45/2)
=cos45
=square root of 2/2

Example 8:
Simplify the given expression.
cos^2(pi/12)-sin^2(pi/12)
=cos2(pi/12)
=cos2(15)
=cos30degrees
=square root of 3/2

Example 9:
angleA is an acute angle.
If sinA=5/13, find sin2A and cos2A
x^2+5^2=13^2
x^2+25=169
x^2=144
x=12
cosA=12/13 tanA=5/12
sin2A=2(5/13)(12/13)
sin2A=120/169
cos2A=1-2(5/13)^2
=1-2(25/169)
=1-50/169
cos2A=119/169

Example 10:
angleA is an acute angle.
If tanA=4/2, find cos2A and tan2A
2^2+1^2=r^2
4+1=r^2
5=r^2
r=square root of 5
sinA=square root of 5/5 cosA=2square root of 5/5
cos2A=1-2(square root of 5/5)^2
=1-2(5/25)=1-10/25
=15/25
cos2A=3/5
tan2A=2(1/2)/1-(1/2)^2=2/2=1/1-2/4
=1/1-1/2=1/1/2
tan2A=2

10-1

sin(30degrees+theta)+sin(30degrees-theta)
sin30degrees*cos theta+cos30degrees*sin theta+sin30degrees*cos theta-cos30degrees*sin theta
(1/2)cos theta+(square root of 3/2)sin theta+(1/2)cos theta-square root of 3/2)sin theta
1cos theta=cos theta

To solve this problem, you have to look at the formulas and find the formula that matches with this problem.  Sin(theta+/-beta)=sin theta*cos beta+/-cos theta*sin beta is the formula you need for this problem.  Once you figure out what formula to use, you plug what you have into the formula.  Then you find the exact values of what you have.  Look on your trig chart for the exact value of sin30 and cos30.  Sin30 is 1/2 and cos30 is square root of 3/2.  Then, you can cancel the +square root of 3/2sin theta and -square root of 3/2sin theta out.  Now all you have left is (1/2)cos theta+(1/2)cos theta.  1/2+1/2 is 1 and you keep the cos theta.  So 1cos theta equals cos theta.

In 10-3 we have a lot of formulas to remember.
1. sin2(alpha) = 2sin(alpha)cos(beta)
2. cos2(alpha) = cos^2(alpha)-sin^2(alpha)
3. cos2(alpha) = 1-2sin^2(alpha)
4. cos2(alpha) = 2cos^2(alpha)-1
5. tan2(alpha) = 2tan(alpha)/1-tan^2(alpha)
6.sin(alpha)/2 = +or- square root of 1-cos(alpha)/2
7. cos(alpha)/2 = +or- square root of 1+cos(alpha)/2
8. tan(alpha)/2 = +or- square root of 1-cos(alpha)/1+cos(alpha)
9. tan(alpha)/2 = sin(alpha)/1+cos(alpha)
10. tan(alpha)/2 = 1-cos(alpha)/sin(alpha)

When you are solving these types of problems:
1. Look at the problem and figure out which of the formulas you need to solve it. 
2. Fill in "alpha" and "beta" with the numbers that you are given. 
3. Simplify the equation until it can not be simplified any longer. 

10-3 babyy :)

So in 10-3 theres a bagillion(my new word) :D formulas!

1. sin2(alpha) = 2sin(alpha)cos(beta)

2. cos2(alpha) = cos^2(alpha)-sin^2(alpha)

3. cos2(alpha) = 1-2sin^2(alpha)

4. cos2(alpha) = 2cos^2(alpha)-1

5. tan2(alpha) = 2tan(alpha)/1-tan^2(alpha)

6.sin(alpha)/2 = +or- square root of 1-cos(alpha)/2

7. cos(alpha)/2 = +or- square root of 1+cos(alpha)/2

8. tan(alpha)/2 = +or- square root of 1-cos(alpha)/1+cos(alpha)

9. tan(alpha)/2 = sin(alpha)/1+cos(alpha)

10. tan(alpha)/2 = 1-cos(alpha)/sin(alpha)

Simplify.

Ex. 1.) cos 2 (x^2) = 2x^4-1

In 10-3 we have a lot of formulas to remember.
1. sin2(alpha) = 2sin(alpha)cos(beta)
2. cos2(alpha) = cos^2(alpha)-sin^2(alpha)
3. cos2(alpha) = 1-2sin^2(alpha)
4. cos2(alpha) = 2cos^2(alpha)-1
5. tan2(alpha) = 2tan(alpha)/1-tan^2(alpha)
6.sin(alpha)/2 = +or- square root of 1-cos(alpha)/2
7. cos(alpha)/2 = +or- square root of 1+cos(alpha)/2
8. tan(alpha)/2 = +or- square root of 1-cos(alpha)/1+cos(alpha)
9. tan(alpha)/2 = sin(alpha)/1+cos(alpha)
10. tan(alpha)/2 = 1-cos(alpha)/sin(alpha)

When you are solving these types of problems:
1. Look at the problem and figure out which of the formulas you need to solve it.
2. Fill in "alpha" and "beta" with the numbers that you are given.
3. Simplify the equation until it can not be simplified any longer.