Precalculus Examples

$\frac{π}{2}$

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 $a = \frac{π}{2}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\frac{π}{2})}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(\frac{π}{2})}^{2}}$

Step 4.2

Apply the product rule to $\frac{π}{2}$ .

$| z | = \sqrt{0 + \frac{π^{2}}{2^{2}}}$

Step 4.3

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

$| z | = \sqrt{0 + \frac{π^{2}}{4}}$

Step 4.4

Add $0$ and $\frac{π^{2}}{4}$ .

$| z | = \sqrt{\frac{π^{2}}{4}}$

Step 4.5

Rewrite $\sqrt{\frac{π^{2}}{4}}$ as $\frac{\sqrt{π^{2}}}{\sqrt{4}}$ .

$| z | = \frac{\sqrt{π^{2}}}{\sqrt{4}}$

Step 4.6

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

$| z | = \frac{π}{\sqrt{4}}$

Step 4.7

Simplify the denominator.

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Step 4.7.1

Rewrite $4$ as $2^{2}$ .

$| z | = \frac{π}{\sqrt{2^{2}}}$

Step 4.7.2

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

$| z | = \frac{π}{2}$

Step 5

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

$θ = \arctan (\frac{0}{\frac{π}{2}})$

Step 6

Substitute the values of $θ = \arctan (\frac{0}{\frac{π}{2}})$ and $| z | = \frac{π}{2}$ .

$\frac{π}{2 (\cos (\arctan (\frac{0}{\frac{π}{2}})) + i \sin (\arctan (\frac{0}{\frac{π}{2}})))}$