Question:
Amplitude of $\frac{1+\sqrt{3}i}{\sqrt{3}+1}$ is :
Amplitude of $\frac{1+\sqrt{3}i}{\sqrt{3}+1}$ is :
Updated On: Apr 28, 2024
- $\frac{\pi}{6}$
- $\frac{\pi}{4}$
- $\frac{\pi}{3}$
- $\frac{\pi}{2}$
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The Correct Option is C
Solution and Explanation
Let $r\left(cos\,\theta+i\,sin\,\theta\right)$
$= \frac{1+i\sqrt{3}}{\sqrt{3}+1} =\frac{1}{\sqrt{3}+1}+i\frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow r\,cos\,\theta = \frac{1}{\sqrt{3}+1} ; r\,sin\,\theta = \frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow tan\, \theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3}.$
$= \frac{1+i\sqrt{3}}{\sqrt{3}+1} =\frac{1}{\sqrt{3}+1}+i\frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow r\,cos\,\theta = \frac{1}{\sqrt{3}+1} ; r\,sin\,\theta = \frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow tan\, \theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3}.$
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.