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Question: Find real part of\(\cos^{- 1}\left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)\)...

Find real part ofcos⁡−1(32+i2)\cos^{- 1}\left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)

A

π3\frac{\pi}{3}

B

π4\frac{\pi}{4}

C

log(3−12)\left( \frac{\sqrt{3} - 1}{2} \right)

D

None

Answer

π4\frac{\pi}{4}

Explanation

Solution

∵\because Expression

cos⁡−1(cos⁡θ+isin⁡θ)=sin⁡−1sin⁡θ−ilog⁡(sin⁡θ+1+sin⁡θ)\cos^{- 1}(\cos\theta + i\sin\theta) = \sin^{- 1}\sqrt{\sin\theta} - i\log(\sqrt{\sin\theta} + \sqrt{1 + \sin\theta}) Where θ=π6\theta = \frac{\pi}{6}

∴\therefore cos⁡−1(32+i2)=\cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } + \frac { i } { 2 } \right) = sin⁡−112−ilog⁡(12+1+12)\sin^{- 1}\sqrt{\frac{1}{2}} - i\log\left( \sqrt{\frac{1}{2}} + \sqrt{1 + \frac{1}{2}} \right)

=π4−ilog⁡(1+32)\frac{\pi}{4} - i\log\left( \frac{1 + \sqrt{3}}{2} \right)=π4+ilog⁡(3−12)\frac{\pi}{4} + i\log\left( \frac{\sqrt{3} - 1}{2} \right)

Real part = π4,\frac{\pi}{4}, Imaginary part = log⁡(3−12)\log\left( \frac{\sqrt{3} - 1}{2} \right)