Write the complex number in trigonometric form

defazajx
2021-02-15
Answered

Write the complex number in trigonometric form

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pierretteA

Answered 2021-02-16
Author has **102** answers

The trigonometric form of the complex number is $r(\mathrm{cos}\theta +i\mathrm{sin}\theta ).$

Here, r is the modulus of the complex number and theta is called the argument of the complex number.

The formula for the modulus of any complex number a+bi is defined as,$r=\sqrt{{a}^{2}+{b}^{2}}$

Substitute$a=5\sqrt{3},b=5$ in the formula for modulus.

$r=\sqrt{{\left(5\sqrt{3}\right)}^{2}+{5}^{2}=\sqrt{75+25}=\sqrt{100}=10}$

Since r is the modulus, it is always positive. So, r= 10.

The formula for the argument of the complex number is defined as,$\theta ={\mathrm{tan}}^{-1}\left(\frac{b}{a}\right)$

Substitute$a=5\sqrt{3},b=5$ in the formula for argument

$\theta ={\mathrm{tan}}^{-1}\left(\frac{5}{5\sqrt{3}}\right)={\mathrm{tan}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

The value of tan is$\frac{1}{\sqrt{3}}$ for $\frac{\pi}{6}$

So,$\theta =\frac{\pi}{6}$

So, the trigonometric form of the given complex number is$10(\mathrm{cos}\left(\frac{\pi}{6}\right)+i\mathrm{sin}\left(\frac{\pi}{6}\right)$

Here, r is the modulus of the complex number and theta is called the argument of the complex number.

The formula for the modulus of any complex number a+bi is defined as,

Substitute

Since r is the modulus, it is always positive. So, r= 10.

The formula for the argument of the complex number is defined as,

Substitute

The value of tan is

So,

So, the trigonometric form of the given complex number is

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