Find an exact value. cosine of negative seven pi divided by twelve

We want to find cos (-7π/12).

Let x = - (7π) / 12.

Then 2x = - (7π) / 6.

Note that

cos (2x) = 2cos²x - 1, or

cos x = + / - √ ([1 + cos (2x) ]/2)

Because - (7π) / 6 is in the 3rd quadrant, with a reference angle of π/6.

cos (-7π/6) = - cos (π/6) = - √3/2 = - 0.866.

Also, - (7π) / 12 is in the 2nd quadrant, therefore it's cosine is negative.

Therefore, obtain

cos (-7π/12) = - √[0.5 * (1 - 0.866) ] = - 0.2588.

Answer: - 0.2588