Question

Use the sum and difference identities to find the exact values of the sine, cosine, and tangent of the angle. . 165 = 135 + 30

Answer 1

So lets get to the problem

165°= 135° +30°

To make it easier I'm going to write the same thing like this
165°= 90° + 45°+30°

Sin165°
= Sin ( 90° + 45°+30° )
= Cos( 45°+30° )..... (∵ Sin(90 + θ)=cosθ
= Cos45°Cos30° - Sin45°Sin30°

Cos165°
= Cos ( 90° + 45°+30° )
= -Sin( 45°+30° )..... (∵Cos(90 + θ)=-Sinθ
= Sin45°Cos30° + Cos45°Sin30°


Tan165°
= Tan ( 90° + 45°+30° )
= -Cot( 45°+30° )..... (∵Cot(90 + θ)=-Tanθ
= -1/tan(45°+30°)
= -[1-tan45°.Tan30°]/[tan45°+Tan30°]

Substitute the above values with the following... These should be memorized
Sin 30° = 1/2
Cos 30° =[Sqrt(3)]/2
Tan 30° = 1/[Sqrt(3)]
Sin45°=Cos45°=1/[Sqrt(2)]
Tan 45° = 1