Question

How do you use the sum and difference formula to simplify `sin165`?

Answer 1

Use sin(135+30).

Explanation:

The sum formula for sin is `sin(A+B)=sinAcosB+cosAsinB`

Choose `A=135` and `B=30` because both of these values lie on the unit circle. From the unit circle:
`cos135=-sqrt2/2`
`sin135=sqrt2/2`
`cos30=sqrt3/2`
`sin30=1/2`

Let `A=135` and `B=30`
Substitute.
`sin(165)=sin(135+30)=sqrt2/2*sqrt3/2 + -sqrt2/2 * 1/2`

`sin(165)= sqrt6/4 -sqrt2/4`
`sin(165)=(sqrt6-sqrt2)/4`