Find the modulus and the argument of the complex number $z=-\sqrt{3}+i$
Find the modulus and the argument of the complex number $z=-\sqrt{3}+i$
$z=-\sqrt{3}+i$
Let $r \cos \theta=-\sqrt{3}$ and $r \sin \theta=1$
On squaring and adding, we obtain
$r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(-\sqrt{3})^{2}+1^{2}$
$\Rightarrow r^{2}=3+1=4 \quad\left[\cos ^{2} \theta+\sin ^{2} \theta=1\right]$
$\Rightarrow r=\sqrt{4}=2 \quad[\text { Conventionally }, r>0]$
$\therefore$ Modulus $=2$
$\therefore 2 \cos \theta=-\sqrt{3}$ and $2 \sin \theta=1$
$\Rightarrow \cos \theta=\frac{-\sqrt{3}}{2}$ and $\sin \theta=\frac{1}{2}$
$\therefore \theta=\pi-\frac{\pi}{6}=\frac{5 \pi}{6}$ [As $\theta$ lies in the $II$ quadrant]
Thus, the modulus an argument of the complex number $-\sqrt{3}+i$ are $2$ and $\frac{5 \pi}{6}$ respectively.
Similar Questions
If $z_1 , z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers, then $\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals
If $z_1 , z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers, then $\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals
- [JEE MAIN 2014]
If complex numbers $z_1$, $z_2$ are such that $\left| {{z_1}} \right| = \sqrt 2 ,\left| {{z_2}} \right| = \sqrt 3$ and $\left| {{z_1} + {z_2}} \right| = \sqrt {5 - 2\sqrt 3 }$, then the value of $|Arg z_1 -Arg z_2|$ is
If complex numbers $z_1$, $z_2$ are such that $\left| {{z_1}} \right| = \sqrt 2 ,\left| {{z_2}} \right| = \sqrt 3$ and $\left| {{z_1} + {z_2}} \right| = \sqrt {5 - 2\sqrt 3 }$, then the value of $|Arg z_1 -Arg z_2|$ is
If $z_1, z_2 $ are any two complex numbers, then $|{z_1} + \sqrt {z_1^2 - z_2^2} |$ $ + |{z_1} - \sqrt {z_1^2 - z_2^2} |$ is equal to
If $z_1, z_2 $ are any two complex numbers, then $|{z_1} + \sqrt {z_1^2 - z_2^2} |$ $ + |{z_1} - \sqrt {z_1^2 - z_2^2} |$ is equal to
If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to
If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to
If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z - 7 - 9i|$ is equal to
If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z - 7 - 9i|$ is equal to
- [IIT 1990]