Precalculus Examples

$3 + 2 i$

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 $a = 3$ and $b = 2$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

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

$| z | = \sqrt{4 + 9}$

Step 4.3

Add $4$ and $9$ .

$| z | = \sqrt{13}$

Step 5

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

$θ = \arctan (\frac{2}{3})$

Step 6

Since inverse tangent of $\frac{2}{3}$ produces an angle in the first quadrant, the value of the angle is $0.5880026$ .

$θ = 0.5880026$

Step 7

Substitute the values of $θ = 0.5880026$ and $| z | = \sqrt{13}$ .

$\sqrt{13} (\cos (0.5880026) + i \sin (0.5880026))$