Question

How do you use the sum and difference identities to find the exact value of `cos 15^@`?

Answer 1

The special triangles, 30-60-90 and 45-45-90, allow us to evaluate sin and cos.

We leverage that information to evaluate `cos(15)`.

`cos(a-b)=cos(a)cos(b)+sin(a)sin(b)`

`cos(60-45)=cos(60)cos(45)+sin(60)sin(45)`

`cos(15)=cos(60)cos(45)+sin(60)sin(45)`

`cos(15)=1/2*1/sqrt(2)+sqrt(3)/2*1/sqrt(2)`

`cos(15)=1/(2sqrt(2))+sqrt(3)/(2sqrt(2))`

`cos(15)=(1+sqrt(3))/(2sqrt(2))`

`cos(15)=(1+sqrt(3))/(2sqrt(2))*sqrt(2)/sqrt(2)`

`cos(15)=(sqrt(2)+sqrt(6))/4`

`cos(15)=0.9659258263`

Verify your results using a calculator. Make sure the calculator is in Degree Mode.