Precalculus Examples

${(3 c i s \frac{π}{6})}^{3}$

Step 1

Cancel the common factor of $3$ .

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Step 1.1

Factor $3$ out of $3 c i s$ .

${(3 (c i s) \frac{π}{6})}^{3}$

Step 1.2

Factor $3$ out of $6$ .

${(3 (c i s) \frac{π}{3 (2)})}^{3}$

Step 1.3

Cancel the common factor.

${(3 (c i s) \frac{π}{3 \cdot 2})}^{3}$

Step 1.4

Rewrite the expression.

${(c i s \frac{π}{2})}^{3}$

Step 2

Combine $c$ and $\frac{π}{2}$ .

${(i s \frac{c π}{2})}^{3}$

Step 3

Combine $i$ and $\frac{c π}{2}$ .

${(s \frac{i (c π)}{2})}^{3}$

Step 4

Combine $s$ and $\frac{i (c π)}{2}$ .

${(\frac{s (i (c π))}{2})}^{3}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

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

$\frac{{(s i c π)}^{3}}{2^{3}}$

Step 5.2

Apply the product rule to $s i c π$ .

$\frac{{(s i c)}^{3} π^{3}}{2^{3}}$

Step 5.3

Apply the product rule to $s i c$ .

$\frac{{(s i)}^{3} c^{3} π^{3}}{2^{3}}$

Step 5.4

Apply the product rule to $s i$ .

$\frac{s^{3} i^{3} c^{3} π^{3}}{2^{3}}$

Step 6

Simplify the numerator.

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Step 6.1

Factor out $i^{2}$ .

$\frac{s^{3} (i^{2} \cdot i) c^{3} π^{3}}{2^{3}}$

Step 6.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{s^{3} (- 1 \cdot i) c^{3} π^{3}}{2^{3}}$

Step 6.3

Rewrite $- 1 i$ as $- i$ .

$\frac{s^{3} \cdot - 1 i c^{3} π^{3}}{2^{3}}$

Step 7

Simplify the expression.

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Step 7.1

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

$\frac{s^{3} \cdot - 1 i c^{3} π^{3}}{8}$

Step 7.2

Move $- 1$ to the left of $s^{3}$ .

$\frac{- 1 \cdot s^{3} i c^{3} π^{3}}{8}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{s^{3} i c^{3} π^{3}}{8}$

Step 8

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 9

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 10

Substitute the actual values of $a = 0$ and $b = - \frac{1 s^{3} c^{3} π^{3}}{8}$ .

$| z | = \sqrt{{(- \frac{1 s^{3} c^{3} π^{3}}{8})}^{2}}$

Step 11

Find $| z |$ .

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Step 11.1

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

$| z | = \frac{1 s^{3} c^{3} π^{3}}{8}$

Step 11.2

Multiply $s^{3}$ by $1$ .

$| z | = \frac{s^{3} c^{3} π^{3}}{8}$

Step 12

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

$θ = \arctan (\frac{- \frac{1 s^{3} c^{3} π^{3}}{8}}{0})$

Step 13

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 14

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = \frac{s^{3} c^{3} π^{3}}{8}$ .

$\frac{s^{3} c^{3} π^{3}}{8 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))}$