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#