Question

If $\sqrt 3 + i = r\left( {\cos \theta + i\sin \theta } \right)$. Find the value of $\theta $.

Answer 1

Hint:
We have given in the question that $\sqrt 3 + i = r\left( {\cos \theta + i\sin \theta } \right)$
Then, we have to find the value of $\theta $.
First, we have to compare the real and imaginary part of the equation
After that we have to find the value of r and from that we get the value of $\theta $.

Complete step by step solution:
We have given in the question that $\sqrt 3 + i = r\left( {\cos \theta + i\sin \theta } \right)$
Then, we have to find the value of $\theta $ .
Now,
It is given that $\sqrt 3 + i = r\left( {\cos \theta + i\sin \theta } \right)$
Then,
Compare the real part and imaginary part of the equation
 $r\cos \theta = \sqrt 3 $ (I)
 $r\sin \theta = 1$ (II)
Now, squaring on both the side
 ${r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta = 3 + 1$
Take out ${r^2}$ common
 $\Rightarrow {r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) = 4$
As, we know that $\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) = 1$
 $\Rightarrow {r^2}\left( 1 \right) = 4$
 $\Rightarrow {r^2} = 4$
 $\Rightarrow r = 2$
Now, put the value of r in the equation (I) and (II)
 $\because 2\cos \theta = \sqrt 3 $
 $\Rightarrow \cos \theta = \dfrac{{\sqrt 3 }}{2}$ (III)
 $\because 2\sin \theta = 1$
 $\Rightarrow \sin \theta = \dfrac{1}{2}$ (IV)
 $\Rightarrow $ From equation (III) and (IV)

$\theta = \dfrac{\pi }{6}$

Note:
Polar form: The polar form of a complex number is another way of representing a complex number.
The form $z = a + ib$ is called the rectangular coordinates form of a complex number. The horizontal axis is the real axis and the vertical axis is the imaginary axis.
 $z = a + ib$
 $z = r\cos \theta + \left( {r\sin \theta } \right)i$
 $z = r\left( {\cos \theta + i\sin \theta } \right)$