Question

How do you use the sum or difference identities to find the exact value of `cos105^circ`?

Answer 1

`- (sqrt(2 - sqrt3))/2`

Explanation:

Use trig identity:
`2cos^2 a = 1 + cos 2a`
`2cos^2 (105) = 1 + cos (210)`. Find cos (210)
Use trig table and unit circle
`cos (210) = cos (30 + 180) = - cos 30 = - sqrt3/2`
`2cos^2 (105) = 1 - sqrt3/2 = (2 - sqrt3)/2`
`cos^2 (105) = (2 - sqrt3)/4`
`cos (105) = +- sqrt(2 - sqrt3)/2`
Since cos 105 (Quadrant II) is negative, take the negative value as answer.