Precalculus Examples

$\sqrt{6} (\cos (315 °) + i \sin (315 °))$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sqrt{6} (\cos (45) + i \sin (315 °))$

Step 1.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{6} (\frac{\sqrt{2}}{2} + i \sin (315 °))$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$\sqrt{6} (\frac{\sqrt{2}}{2} + i (- \sin (45)))$

Step 1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{6} (\frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2}))$

Step 1.5

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$\sqrt{6} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})$

Step 2

Apply the distributive property.

$\sqrt{6} \frac{\sqrt{2}}{2} + \sqrt{6} (- \frac{i \sqrt{2}}{2})$

Step 3

Multiply $\sqrt{6} \frac{\sqrt{2}}{2}$ .

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Step 3.1

Combine $\sqrt{6}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} \sqrt{2}}{2} + \sqrt{6} (- \frac{i \sqrt{2}}{2})$

Step 3.2

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2}}{2} + \sqrt{6} (- \frac{i \sqrt{2}}{2})$

Step 3.3

Multiply $6$ by $2$ .

$\frac{\sqrt{12}}{2} + \sqrt{6} (- \frac{i \sqrt{2}}{2})$

Step 4

Multiply $\sqrt{6} (- \frac{i \sqrt{2}}{2})$ .

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Step 4.1

Combine $\sqrt{6}$ and $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{12}}{2} - \frac{\sqrt{6} (i \sqrt{2})}{2}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt{12}}{2} - \frac{\sqrt{2 \cdot 6} i}{2}$

Step 4.3

Multiply $2$ by $6$ .

$\frac{\sqrt{12}}{2} - \frac{\sqrt{12} i}{2}$

Step 5

Simplify each term.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.1.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)}}{2} - \frac{\sqrt{12} i}{2}$

Step 5.1.1.2

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

$\frac{\sqrt{2^{2} \cdot 3}}{2} - \frac{\sqrt{12} i}{2}$

Step 5.1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{3}}{2} - \frac{\sqrt{12} i}{2}$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{2} - \frac{\sqrt{12} i}{2}$

Step 5.2.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3} - \frac{\sqrt{12} i}{2}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.3.1.1

Factor $4$ out of $12$ .

$\sqrt{3} - \frac{\sqrt{4 (3)} i}{2}$

Step 5.3.1.2

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

$\sqrt{3} - \frac{\sqrt{2^{2} \cdot 3} i}{2}$

Step 5.3.2

Pull terms out from under the radical.

$\sqrt{3} - \frac{2 \sqrt{3} i}{2}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\sqrt{3} - \frac{2 \sqrt{3} i}{2}$

Step 5.4.2

Divide $\sqrt{3} i$ by $1$ .

$\sqrt{3} - (\sqrt{3} i)$

$\sqrt{3} - \sqrt{3} i$