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Inverse Trigonometric Functions

Question
CBSEENMA12036008
Wired Faculty App

A value of θ for which fraction numerator 2 space plus space 3 straight i space sin space straight theta over denominator 1 minus 2 straight i space sin space straight theta end fraction is purely imaginary is

  • π/3

  • π/6

  • sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 4 end fraction close parentheses
  • sin to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses

Solution

D.

sin to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses

Let z = straight z space equals space fraction numerator 2 space plus space 3 straight i space sin space straight theta over denominator 1 minus 2 straight i space sin space straight theta end fraction is purely imaginary. Then, we have Re (z) = 0
We have Re (z) = 0
Now, consider z = 
straight z equals space fraction numerator 2 space plus space 3 straight i space sin space straight theta over denominator 1 minus 2 straight i space sin space straight theta end fraction space equals space fraction numerator left parenthesis 2 plus space 3 straight i space sin space straight theta right parenthesis left parenthesis 1 space plus space 2 straight i space sin space straight theta right parenthesis over denominator left parenthesis 1 minus 2 straight i space sin space straight theta right parenthesis left parenthesis 1 plus space 2 straight i space sin space straight theta right parenthesis end fraction equals space fraction numerator 2 space plus space 4 straight i space sin space straight theta space plus space 3 straight i space sin space straight theta space plus space 6 space straight i squared space sin space squared space straight theta over denominator 1 squared minus space left parenthesis 2 straight i space sin space straight theta right parenthesis squared end fraction equals space fraction numerator 2 space plus space 7 space straight i space sin space straight theta space minus space 6 space sin squared space straight theta over denominator 1 space plus space 4 space sin squared space straight theta end fraction equals space fraction numerator 2 minus 6 space sin squared space straight theta over denominator 1 plus 4 space sin squared space straight theta end fraction space plus space straight i fraction numerator 7 space sin space straight theta over denominator 1 space plus space 4 space sin squared space straight theta end fraction therefore space Re space left parenthesis straight z right parenthesis space equals space 0 therefore space fraction numerator 2 minus 6 space sin squared space straight theta over denominator 1 plus space 4 space sin squared space straight theta end fraction space equals space 0 rightwards double arrow space 2 space equals space 6 space sin squared straight theta rightwards double arrow space sin squared straight theta space equals space 1 third rightwards double arrow space sin space straight theta space equals space plus-or-minus space fraction numerator 1 over denominator square root of 3 end fraction rightwards double arrow space straight theta space equals space sin to the power of negative 1 end exponent space open parentheses plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction close parentheses space equals space plus-or-minus space sin to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses

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