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Argument of complex number - (Concept)
If a complex number z = x + iy is represented by a point P in the Argand plane and OP forms some angle with a positive x-axis, let's denote it with ?, then ? is called the argument of z.
$\begin{aligned} & \tan \theta=\frac{\mathrm{PM}}{\mathrm{OM}} \\ & \tan \theta=\frac{\mathrm{y}}{\mathrm{x}}=\frac{\operatorname{Im}(\mathrm{z})}{\operatorname{Re}(\mathrm{z})} \Rightarrow \theta=\tan ^{-1} \frac{\mathrm{y}}{\mathrm{x}} \\ & \arg (\mathrm{z})=\theta=\tan ^{-1} \frac{\mathrm{y}}{\mathrm{x}}\end{aligned}$
If ? lies between -? < ? ≤ ?, then ? is called the principal argument. The value of the argument differs depending on which quadrant point (x,y) lies.
If it lies in 1st quadrant then it is ? (acute angle)
If the point lies in 2nd quadrant, then $\arg (z)=\theta=\pi-\tan ^{-1} \frac{y}{|x|}$
So it will be an obtuse +ve angle
If the point lies in lies in 3rd quadrant then $\arg (z)=\theta=-\pi+\tan ^{-1} \frac{y}{x}$
It will be an obtuse -ve angle
If the point lies in 4th quadrant then $\arg (z)=\theta=-\tan ^{-1} \frac{|y|}{x}$
It will be a -ve acute angle
Note:
If $\arg (\mathrm{z})=\frac{\pi}{2}$ or $-\frac{\pi}{2}, \mathrm{z}$ is purely imaginary.
If $\arg (\mathrm{z})=0$ or $\pi, \mathrm{z}$ is purely real.
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
Arg equals:
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Let be a root of the quadratic equation,
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then arg z is equal to :
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Which of the following is one of the argument of :
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$\text { If } \frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi]
$
is real number, then an argument of $\sin \theta+i \cos \theta$ is :
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Let a complex number be $w=1-\sqrt{3} i$. Let the another complex number $z$ be such that $|z w|=1_{\text {and }} \arg (z)-\arg (w)=\frac{\pi}{2}$. Then the area of the triangle with the vertices origin, $a$ and $w$ is equal to:
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Arg equals:
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Which of the following can't be argument of :
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Principal argument of equals:
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The area of the polygon, whose vertices are the non-real roots of the equation
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Let z satisfy and
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Statement I : z is a real number.
Statement II: Principal argument of z is .
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The point of intersection the curves arg $(z-i+2)=\frac{\pi}{6}$ & arg $(z+4-3 i)=-\frac{\pi}{4}$ is given by
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Let and
Then
is equal to
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Let w1 be the point obtained by the rotation of z1 = 5 + 4i about the origin through a right angle in the
anticlockwise direction, and w2 be the point obtained by the rotation of z2 = 3 + 5i about the origin through a right angle in the clockwise direction. Then the principal argument of w1 – w2 is equal to :
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If z1, z2 and z3, z4 are 2 pairs of complex conjugate numbers, thenequals:
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If general argument of a complex no is $2n\pi +\frac{4\pi }{3};$ what is the principle argument
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The argument of the complex no. $\frac{2+3 i}{3+i+(1+2 i)^2}$ is
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Principal value of amplitude of $(1+i)$ is:
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Principal value of the argument of $\cos 1200^{o}+i \sin 1200^{o}$ is:
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The point represented by the complex number $2 -i$ is rotated about origin through an angle $\pi/2$ in the clockwise direction, the new position of point is:
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What is the argument of zw , such that $\arg (z)=\frac{2 \pi}{3}$ and $\arg (w)=\frac{\pi}{2}$.
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Area of the whole circle of which the locus of the points on a major arc is given by $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ is given by $A$. Then the greatest integral value of $A$ is________.
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The value of $\arg (x)$ when $x<0$is:
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The principle argument of $Z= -1+\sqrt{3}i$ is
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In the complex plane, let $z_1=\sqrt{3}+i$ and $z_2=\sqrt{3}-i$ be two adjacent vertices of an $n$ - sided regular polygon centered at the origin. Then $n$ equals
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Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2 \sqrt{2} i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}$. Then
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Find principal argument of $\left(\frac{Z_1}{Z_2}\right)$, where $\operatorname{Arg}\left(Z_1\right)=-\frac{2 \pi}{3} \operatorname{Arg}\left(Z_2\right)=+\frac{\pi}{2}$.
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Let $\mathrm{A}=$
$\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+i \sin \theta}{\cos \theta-3 i \sin \theta}\right)=0\right\} .$
Then $\sum_{\theta \in A} \theta^2$ is equal to
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The values of real parameter $\alpha$ for which $\left|z-\left(\alpha^2-7 \alpha+11+i\right)\right|=1$ and $\arg z \geqslant \frac{\pi}{2}$ is Satisfied for atleast one z, are:
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Polar representation of a complex number
Let the point P represent the nonzero complex number z = x + iy. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis
