Question:
The principal argument of the complex numb $Z=\frac{1+\sin \frac{\pi}{3}+i \cos\frac{\pi}{3} }{1+\sin \frac{\pi}{3} - i \cos\frac{\pi}{3} }$ is
The principal argument of the complex numb $Z=\frac{1+\sin \frac{\pi}{3}+i \cos\frac{\pi}{3} }{1+\sin \frac{\pi}{3} - i \cos\frac{\pi}{3} }$ is
Updated On: Jun 8, 2024
- $\frac{\pi}{ 3}$
- $\frac{\pi}{6 }$
- $\frac{2\pi}{ 3}$
- $\frac{\pi}{2 }$
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The Correct Option is B
Solution and Explanation
$\arg (z)=\arg ($ Numerator $)-\arg ($ Denominator $)$
$=\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|+\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|$
$=2 \tan ^{-1}\left[\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right]$
$=2 \tan ^{-1}(2-\sqrt{3})$
$=2 \times 15^{\circ}=30^{\circ}=\frac{\pi}{6}$
$=\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|+\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|$
$=2 \tan ^{-1}\left[\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right]$
$=2 \tan ^{-1}(2-\sqrt{3})$
$=2 \times 15^{\circ}=30^{\circ}=\frac{\pi}{6}$
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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.