# How do you evaluate cos^-1(-sin(2/3 pi))?

Feb 11, 2016

cos^(-1)(−sin(2/3π))=7π/6

#### Explanation:

cos^(-1)(−sin(2/3π) means the angle whose cosine is −sin(2/3π).

If $\theta$ is such an angle, it is apparent that

cos theta=−sin(2/3π) - Equation $\left(A\right)$

Also referring to the identity cos theta = - cos (π-theta)

and cos alpha= sin (π/2 - alpha), we can write

cos theta=−cos (π-theta)=- sin(π/2 - (π-theta))

or cos theta = - sin(-π/2 +theta) - Equation $\left(B\right)$

From equations A and B it is apparent that

2/3π=(-π/2 +theta) or

theta=(2/3π+π/2)=7π/6

Hence cos^(-1)(−sin(2/3π))=7π/6