Question

How do you find the exact functional value cos 165 using the cosine sum or difference identity?

Answer 1

Find cos (165)

Ans: `- sqrt(2 + sqrt3)/2`

Explanation:

Call cos 165 = cos t
`cos 2t = cos 330 = cos (150 + 180) = - cos 150 = sqrt3/2.`
Apply the trig identity: `cos 2t = 2cos^2 t - 1`
`sqrt3/2 = 2cos^2 t - 1`
`cos^2 t = (2 + sqrt3)/4` --> `cos t = +- sqrt(2 + sqrt3)/2`
Since the arc 165 is in Quadrant II, its cos is negative, then,
`cos t = cos 165 = -sqrt(2 + sqrt3)/2`

Check by calculator:
cos (-165) = -0.97
`-sqrt(2 = sqrt3)/2 = 0.97.`OK