Inverse Trigonometric Functions

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Using principle value, evaluate the following : - Wired Faculty

Question

Using principle value, evaluate the following :

$cos to the power of negative 1 end exponent open parentheses cos fraction numerator 2 straight pi over denominator 3 end fraction close parentheses plus sin to the power of negative 1 end exponent open parentheses sin space fraction numerator 2 straight pi over denominator 3 end fraction close parentheses$

Solution

Consider space cos to the power of negative 1 end exponent open parentheses cos fraction numerator 2 straight pi over denominator 3 end fraction close parentheses plus sin to the power of negative 1 end exponent open parentheses sin space fraction numerator 2 straight pi over denominator 3 end fraction close parentheses equals cos to the power of negative 1 end exponent open square brackets cos fraction numerator 2 straight pi over denominator 3 end fraction close square brackets plus sin to the power of negative 1 end exponent open square brackets sin open parentheses straight pi minus straight pi over 3 close parentheses close square brackets space space space space space space space space space space space space space space space space space space space space space space space equals cos to the power of negative 1 end exponent open parentheses cos fraction numerator 2 straight pi over denominator 3 end fraction close parentheses plus sin to the power of negative 1 end exponent open parentheses sin space fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 straight pi over denominator 3 end fraction plus straight pi over 3 equals straight pi