Question

How do you find the exact values of the sine, cosine, and tangent of the angle `-165^circ`?

Answer 1

`tan (-165) = (sqrt(2 - sqrt3))/(sqrt(2 + sqrt3)`

Explanation:

sin (-165) = sin (195) --> co-terminal.
Call sin (195) = sin t
Use trig table and unit circle:
`cos 2t = cos 390 = cos (30 + 360) = cos 30 = sqrt3/2`
Use trig identity: `2sin^2 t = 1 - cos 2t`
`2sin^2 t = 1 - sqrt3/2 = (2 - sqrt3)/2`
`sin^2 t = (2 - sqrt3)/4`
`sin t = +- (2 - sqrt3)/2`
Since sin (- 165) is negative, then, take the negative answer.
`sin (-165) = sin t = - (2 - sqrt3)/2`
Use trig identity: `2cos^2 t = 1 + cos 2t = (1 = sqrt3/2) = (2 + sqrt3)/2`
`cos^2 t = (2 + sqrt3)/4` --> `cos t = +- sqrt(2 + sqrt3)/2`
Since cos (-165) is negative, take the negative value.
`cos (-165) = cos t = - sqrt(2 + sqrt3)/2`
`tan (-165) = tan t = sin/(cos) = (sqrt(2 - sqrt3))/(sqrt(2 + sqrt3)`