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If | Z | = 2 and Arg ( Z ) = π 4 ,Find Z.

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Question

If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.

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Solution

We know that,

\[z = \left| z \right|\left\{ cos\left[ \arg\left( z \right) \right] + i\sin\left[ \arg\left( z \right) \right] \right\}\] 
 
\[     = 2\left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right)\] 
 
\[     = 2\left( \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \right)\] 
 
\[     = \sqrt{2}\left( 1 + i \right)\]
Hence, 
\[z = \sqrt{2}\left( 1 + i \right)\].
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Chapter 13: Complex Numbers - Exercise 13.5 [Page 63]

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R.D. Sharma Mathematics [English] Class 11
Chapter 13 Complex Numbers
Exercise 13.5 | Q 23 | Page 63

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