Question 6(i-ix), Exercise 1.4

Solutions of Question 6(i-ix) 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: $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$

Solution.

$$\begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right) \\ =& \sqrt{2} \left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}} \right) \\ =& 1-i. \end{align}$$

Write a given complex number in the algebraic form: $5\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)$

Solution.

$$\begin{align*} &5\left(\cos 210^\circ + i \sin 210^\circ\right) \\ =& 5\left(-\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) \\ =& -\frac{5\sqrt{3}}{2} - \frac{5}{2}i \end{align*}$$

Write a given complex number in the algebraic form: $2\left(\cos \dfrac{3 \pi}{2}+i \sin \dfrac{3 \pi}{2}\right)$

Solution.

$$\begin{align*} &2\left(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}\right) \\ =& 2\left(0 + i(-1)\right) \\ =& -2i \end{align*}$$

Write a given complex number in the algebraic form: $4\left(\cos \dfrac{5 \pi}{6}+i \sin \dfrac{5 \pi}{6}\right)$

Solution.

$$\begin{align*} &4\left(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}\right) \\ =& 4\left(-\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right) \\ =& -2\sqrt{3} + 2i \end{align*}$$

Write a given complex number in the algebraic form: $2\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)$

Solution.

$$\begin{align*} & 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right) \\ &= 2\left(\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right) \\ &= \sqrt{3} + i \end{align*}$$

Write a given complex number in the algebraic form: $\cos 135^{\circ}+i \sin 135^{\circ}$

Solution.

$$\begin{align*} &\cos 135^\circ + i \sin 135^\circ \\ &= -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \end{align*}$$

Write a given complex number in the algebraic form: $10\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)$

Solution.

$$\begin{align*} &10\left(\cos 50^\circ + i \sin 50^\circ\right)\\ &\approx 10\left(0.643 + i 0.766 \right) \\ &= 6.43 + 7.66i \end{align*}$$

Note: Generally, we write exact answers, not approximate answers.

Write a given complex number in the algebraic form: $\sqrt{2}\left(\cos \dfrac{3 \pi}{4}+i \sin \dfrac{3 \pi}{4}\right)$

Solution.

$$\begin{align*} &\sqrt{2}\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\\ &= \sqrt{2}\left(-\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right) \\ &= -1 + i. \end{align*}$$

Write a given complex number in the algebraic form: $4\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right)$

Solution.

$$\begin{align*} &4\left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right) \\ &= 4\left(-\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2}\right) \\ &= -2 + 2\sqrt{3}i. \end{align*}$$

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