Inverse Trigonometric Functions

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Question

Write the principal value of $tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis plus cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses.$

Solution

Let space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis space equals space straight y space space rightwards double arrow space tan space straight y space equals space 1 space equals space tan open parentheses straight pi over 4 close parentheses space rightwards double arrow space space space straight y space equals space straight pi over 4 space rightwards double arrow space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis space equals space straight pi over 4 space cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses space equals space straight z rightwards double arrow space cos space straight z space equals space minus 1 half equals negative cos straight pi over 3 equals cos open parentheses straight pi minus straight pi over 3 close parentheses equals cos open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses rightwards double arrow space straight z space equals space fraction numerator 2 straight pi over denominator 3 end fraction space rightwards double arrow space cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses equals fraction numerator 2 straight pi over denominator 3 end fraction therefore space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis plus cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses equals straight pi over 4 plus fraction numerator 2 straight pi over denominator 3 end fraction equals fraction numerator 11 straight pi over denominator 12 end fraction

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Inverse Trigonometric Functions

Write the principal value of $tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis plus cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses.$

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