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Write (I25)3 in Polar Form. - Mathematics

Write (i25)3 in polar form.

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Solution

\[\left( i^{25} \right)^3 = i^{75} \]

\[ = i^{4 \times 18 + 3} \]

\[ = \left( i^4 \right)^{18} . i^3 \]

\[ = i^3 [ \because i^4 = 1]\]

\[ = - i [ \because i^3 = - i]\]

Let \[z = 0 - i\]

Then, 

\[\left| z \right| = \sqrt{0^2 + \left( - 1 \right)^2} = 1\].

Let θ be the argument of z and α be the acute angle given by 

\[\tan\alpha = \frac{\left| Im\left( z \right) \right|}{\left| Re\left( z \right) \right|}\]
Then, 

\[\tan\alpha = \frac{1}{0} = \infty \]

\[ \Rightarrow \alpha = \frac{\pi}{2}\]

Clearly, z lies in fourth quadrant. So, arg(z) = 

\[- \alpha = - \frac{\pi}{2}\].
∴ the polar form of is  \[\left| z \right|\left( \cos\theta + i\sin\theta \right) = \cos\left( - \frac{\pi}{2} \right) + i\sin\left( - \frac{\pi}{2} \right)\].
Thus, the polar form of  (i25)is \[\cos\left( \frac{\pi}{2} \right) - i\sin\left( \frac{\pi}{2} \right)\].
  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.4 | Q 2 | Page 57
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