====== 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. 

=====Question 6(i)=====
 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}
=====Question 6(ii)=====
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*}
=====Question 6(iii)=====
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*}
=====Question 6(iv)=====
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*}
=====Question 6(v)=====
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*}

=====Question 6(vi)=====
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*}
=====Question 6(vii)=====
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*}

<fc #ff0000>Note:</fc> Generally, we write exact answers, not approximate answers. 
=====Question 6(viii)=====
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*}

=====Question 6(ix)=====
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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