Trigonometry Examples

$- 3 i$

Step 1

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 2

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 3

Substitute the actual values of $a = 0$ and $b = - 3$ .

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

Step 4

Find $| z |$ .

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Step 4.1

Raise $- 3$ to the power of $2$ .

$| z | = \sqrt{9}$

Step 4.2

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2}}$

Step 4.3

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 3$

Step 5

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- 3}{0})$

Step 6

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Step 7

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = 3$ .

$3 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$