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