# Question 6(x-xvii), Exercise 1.4

Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Write a given complex number in the algebraic form: $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$

Solution.

Do yourself as previous parts.

Write a given complex number in the algebraic form: $10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$

Solution.

Do yourself as previous parts.

Write a given complex number in the algebraic form: $2\left(\cos\dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$ Solution.

Do yourself as previous parts.

Write a given complex number in the algebraic form: $\dfrac{1}{\sqrt{2}}\left(\cos \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right)$

Solution.

Do yourself as previous parts.

Write a given complex number in the algebraic form: $7\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$

Solution.

Do yourself as previous parts.

Write a given complex number in the algebraic form: $2 e^{i \dfrac{\pi}{4}}$

Solution.

\begin{align*} & 2 e^{i \frac{\pi}{4}} \\ =& 2\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right) \\ =& 2\left(\frac{1}{\sqrt{2}} + i \cdot \frac{1}{\sqrt{2}}\right) \\ =& \sqrt{2} + \sqrt{2}i. \end{align*}

Write a given complex number in the algebraic form: $3 e^{i \dfrac{\pi}{2}}$

Solution.

\begin{align*} & 3 e^{i \frac{\pi}{2}} \\ =& 3\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right) \\ =& 3\left(0 + i \cdot 1 \right) \\ =& 3i. \end{align*}

Write a given complex number in the algebraic form: $5 e^{i{\dfrac{\pi}{3}}}$.

Solution.

\begin{align*} & 5 e^{i \dfrac{\pi}{3}} \\ =& 5\left(\cos \dfrac{\pi}{3} + i \sin \dfrac{\pi}{3}\right) \\ =& 5\left(\dfrac{1}{2} + i \cdot \dfrac{\sqrt{3}}{2} \right) \\ =& \dfrac{5}{2} + i \cdot \dfrac{5\sqrt{3}}{2}. \end{align*}