double and half angles O.o oooooo

sin2α= 2sinαcosα

cos2α= cos²α -sin²α

cos2α= 1 -2sin²α

cos2α= 2cos²α -1

tan2α= 2tanα/1 -tan²α

sin(α/2)= ±√((1 - cosα)/2)

cos(α/2)= ± √((1 + cosα)/2)

tan(α/2)= ±√(1 - cosα)/(1+cosα)

tan(α/2)= sinα/(1 + cosα)

tan(α/2)= (1 - cosα)/sinα

the plus or minus things are determined by where the original angle is.

Example 1

Find the exact value of sin120° (Hint... use your trig chart.).

sin120°= sin2(60°)

sin120°= 2sin60°cos60°

sin120°= 2(½)*√(3)/2

sin120° = √(3)/2

Example 2 

Find the exact value of cos15° (Hint... use the trig chart.).

cos15°= cos(30°)/2

cos15°= ± √(1 + cos30°)/2

cos15°= ± √(1+½)/2

cos15°= ± √(3/2)/2

cos15°= ±√3/4

cos15°= ±(√3)/2

cos15°= (√3)/2

Example 3

Simplify as much as possible:

sinΘtanΘ + cos2ΘsecΘ

convert everything to sine and cosine

sinΘ(sinΘ/cosΘ)+cos2Θ(1/cosΘ)

change cos2Θ to cos²Θ -sin²Θ

(sin²Θ/cosΘ) + ((cos²Θ -sin²Θ)/cosΘ))

algebra

(sin²Θ/cosΘ) + cosΘ -(sin²Θ/cosΘ)

cancel

cosΘ