Chapter 6, Problem 21STP

To determine

To select:

The correct option of the trigonometric from of the complex number $z=7i$

(A) $\sqrt{8}\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

(B) $7\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

(C) $7\cos \frac{3\pi }{2}$

(D) $\sqrt{8}i\sin \frac{3\pi }{2}$

Answer to Problem 21STP

The correct answer is (B). The trigonometric form of the given complex number $\left(z=7i\right)$ is derived as $7\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$ .

Explanation of Solution

Given information:

  $z=7i$

Calculation:

The modulus of the given complex number $\left(z=0+7i\right)$ is calculated as:

  $r=\sqrt{a^2+b^2}=\sqrt{{\left(0\right)}^2+{\left(7\right)}^2}=\sqrt{49}=7$

The argument of the given complex number $\left(z=0+7i\right)$ is calculated as:

  $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)={\tan }^{-1}\left(\frac{7}{0}\right)={\tan }^{-1}\left(\infty \right)=\frac{\pi }{2}$

Now, the trigonometric form of the given complex number is written as:

  $\begin{array}{l}z=r\left(\cos \theta +i\sin \theta \right)\\ z=7\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)\end{array}$

So, the correct option is (B).