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

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Answer 1 · 2016-02-11T05:44:58

$cos^(-1)(−sin(2/3π))=7π/6$

$cos^(-1)(−sin(2/3π)$ 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 $(A)$

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 $(B)$

From equations A and B it is apparent that

$2/3π=(-π/2 +theta)$ or

$theta=(2/3π+π/2)=7π/6$

Hence $cos^(-1)(−sin(2/3π))=7π/6$

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