Question

How do you use the sum and difference identities to find the exact values of the sine, cosine and tangent of the angle `(7pi)/12`?

Answer 1

Find 3 trig functions of `((7pi)/12)`

Explanation:

Call `sin ((7pi)/12) = sin t` -> `cos 2t = cos ((14pi)/12)`
On the trig unit circle.
`cos ((14pi)/12) = cos (pi/6 + pi) = - cos (pi/6) = -(sqrt3)/2`
Apply the trig identity: `cos 2t = 1 - 2sin^2 t`

`- (sqrt3)/2 = 1 - 2sin^2 t`
`sin^2 t = (2 + sqrt3)/4`
`sin t = sin ((7pi)/12) = +- sqrt(2 + sqrt3)/2`

Apply trig identity: `cos 2t = 2cos^2 t - 1`
`-(sqrt3)/2 = 2cos^2 t - 1`
`2cos^2 t = 1 - (sqrt3)/2 = (2 - sqrt3)/2`
`cos^2 t = (2 - sqrt3)/4`
`cos t = cos ((7pi)/12) = +- sqrt(2 - sqrt3)/2`

`tan ((7pi)/12) = sin t/cos t = +- (sqrt2 +sqrt3)/sqrt(2 - sqrt3)`