Question

State and explain different ways of finding the argument of the complex number.

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Answer 1

Answer:

What Is Argument of Complex Number?

The argument of a complex number is an angle that is inclined from the real axis towards the direction of the complex number which is represented on the complex plane. We can denote it by “θ” or “φ” and can be measured in standard units “radians”.

In the diagram above, the complex number is denoted by the point P. The length OP is the magnitude or modulus of the number, and the angle at which OP is inclined from the positive real axis is known as the argument of the point P.

How To Find Argument Of a Complex Number?

There are few steps that need to be followed if we want to find the argument of a complex number. These steps are given below:

Step 1) First we have to find both real as well as imaginary parts from the complex number that is given to us and denote them x and y respectively.

Step 2) Then we have to use the formula θ = tan−1

(y/x) to substitute the values.

Step 3) If by solving the formula we get a standard value then we have to find the value of θ or else we have to write it in the form of tan−1

itself.

Step 4) The final value along with the unit “radian” is the required value of the complex argument for the given complex number.

With this method you will now know how to find out argument of a complex number.

Argument of Complex Number Examples

Example 1) Find the argument of -1+i and 4-6i

Solution 1) We would first want to find the two complex numbers in the complex plane. This will make it easy for us to determine the quadrants where angles lie and get a rough idea of the size of each angle. For, z= --+i

We can see that the argument of z is a second quadrant angle and the tangent is the ratio of the imaginary part to the real part, in such a case −1. The tangent of the reference angle will thus be 1. Write the value of the second quadrant angle so that its reference angle can have a tangent equal to 1. If the reference angle contains a tangent which is equal to 1 then the value of reference angle will be π/4 and so the second quadrant angle is π − π/4 or 3π/4.

For z = 4 − 6i:

This time the argument of z is a fourth quadrant angle. The reference angle has a tangent 6/4 or 3/2. None of the well known angles consist of tangents with value 3/2. Therefore, the reference angle is the inverse tangent of 3/2, i.e. tan−1

(3/2). Hence the argument being fourth quadrant itself is 2π − tan−1

(3/2).

In order to get a complete idea of the size of this argument, we can use a calculator to compute 2π − tan−1

(3/2) and see that it is approximately 5.3 (radians). In degrees this is about 303o.

Solved Example

Question: Find the argument of a complex number 2 + 23–√

i.

Solution: Let z = 2 + 23–√

i.

The real part, x = 2 and the Imaginary part, y = 23–√

We already know the formula to find the argument of a complex number. That is

arg (z) = tan−1

(y/x)

arg (z) = tan−1

(23–√

/2)

arg (z) = tan−1

(3–√

)

arg (z) = tan−1

(tan π/3)

arg (z) = π/3

Therefore, the argument of the complex number is π/3 radian.

Answer 2

Hope it helps akka

thank you