Find the modulus and argument of the complex nu

Question

Find the modulus and argument of the complex number -√3 + i.

Askiitians Tutor Team · Accepted Answer

To find the modulus and argument of the complex number $- \sqrt{3} + i$ , we proceed as follows:

### Step 1: Find the modulus
The modulus of a complex number $z = x + y i$ is given by:

$| z | = \sqrt{x^{2} + y^{2}}$

Here, $x = - \sqrt{3}$ and $y = 1$ . Substituting these values:

$| z | = \sqrt{(- \sqrt{3})^{2} + 1^{2}}$

$| z | = \sqrt{3 + 1} = \sqrt{4} = 2$

So, the modulus of the complex number is 2.

### Step 2: Find the argument
The argument of a complex number is the angle $θ$ that the line connecting the origin to the point $(- x, y)$ makes with the positive real axis, measured counterclockwise. It is given by:

$θ = \tan^{- 1} (\frac{y}{x})$

Substitute $x = - \sqrt{3}$ and $y = 1$ :

$θ = \tan^{- 1} (\frac{1}{- \sqrt{3}})$

Simplify:

$θ = \tan^{- 1} (- \frac{1}{\sqrt{3}})$

We know that $\tan^{- 1} (- \frac{1}{\sqrt{3}})$ corresponds to an angle in the second quadrant because the real part is negative ( $- \sqrt{3}$ ) and the imaginary part is positive ( $1$ ).

The reference angle for $- \frac{1}{\sqrt{3}}$ is $\frac{π}{6}$ , so the argument is:

$θ = π - \frac{π}{6} = \frac{5 π}{6}$

### Final Answer:
1. **Modulus**: 2
2. **Argument**: $\frac{5 π}{6}$ radians.

11 grade maths others> Find the modulus and argument of the comp...

Find the modulus and argument of the complex number -√3 + i.

Aniket Singh , 5 Months ago

Askiitians Tutor Team

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11 grade maths others> Find the modulus and argument of the comp...

Find the modulus and argument of the complex number -√3 + i.

Aniket Singh , 5 Months ago

Askiitians Tutor Team

LIVE ONLINE CLASSES

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