Precalculus Examples

$6 - 3 i$

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 $a = 6$ and $b = - 3$ .

$| z | = \sqrt{{(- 3)}^{2} + 6^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

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

$| z | = \sqrt{9 + 6^{2}}$

Step 4.2

Raise $6$ to the power of $2$ .

$| z | = \sqrt{9 + 36}$

Step 4.3

Add $9$ and $36$ .

$| z | = \sqrt{45}$

Step 4.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.4.1

Factor $9$ out of $45$ .

$| z | = \sqrt{9 (5)}$

Step 4.4.2

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

$| z | = \sqrt{3^{2} \cdot 5}$

Step 4.5

Pull terms out from under the radical.

$| z | = 3 \sqrt{5}$

Step 5

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

$θ = \arctan (\frac{- 3}{6})$

Step 6

Since inverse tangent of $\frac{- 3}{6}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.4636476$ .

$θ = - 0.4636476$

Step 7

Substitute the values of $θ = - 0.4636476$ and $| z | = 3 \sqrt{5}$ .

$3 \sqrt{5} (\cos (- 0.4636476) + i \sin (- 0.4636476))$