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Question: The argument of the complex number sin\(\frac{6\pi}{5} + i\left( 1 + \cos\frac{6\pi}{5} \right)\) is...

The argument of the complex number sin6π5+i(1+cos⁡6π5)\frac{6\pi}{5} + i\left( 1 + \cos\frac{6\pi}{5} \right) is

A

6π5\frac{6\pi}{5}

B

5π6\frac{5\pi}{6}

C

9π10\frac{9\pi}{10}

D

2π5\frac{2\pi}{5}

Answer

9π10\frac{9\pi}{10}

Explanation

Solution

Sol. sin 6π5+i(1+cos⁡6π5)\frac{6\pi}{5} + i\left( 1 + \cos\frac{6\pi}{5} \right)

=2cos 3π5(sin⁡3π5+icos⁡3π5)\frac{3\pi}{5}\left( \sin\frac{3\pi}{5} + i\cos\frac{3\pi}{5} \right)

=2cos 3π5[cos⁡(−π10)+isin⁡(−π10)]\frac{3\pi}{5}\left\lbrack \cos\left( \frac{- \pi}{10} \right) + i\sin\left( \frac{- \pi}{10} \right) \right\rbrack

=–2cos 3π5[cos⁡9π10+isin⁡9π10]\frac{3\pi}{5}\left\lbrack \cos\frac{9\pi}{10} + i\sin\frac{9\pi}{10} \right\rbrack