Trigonometry Examples

$4 (\cos (30 °) + i \sin (30 °))$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (30 °)$ is $\frac{\sqrt{3}}{2}$ .

$4 (\frac{\sqrt{3}}{2} + i \sin (30 °))$

Step 1.2

The exact value of $\sin (30 °)$ is $\frac{1}{2}$ .

$4 (\frac{\sqrt{3}}{2} + i \frac{1}{2})$

Step 1.3

Combine $i$ and $\frac{1}{2}$ .

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$4 \frac{\sqrt{3}}{2} + 4 \frac{i}{2}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $4$ .

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

Step 2.2.2

Cancel the common factor.

$2 \cdot 2 \frac{\sqrt{3}}{2} + 4 \frac{i}{2}$

Step 2.2.3

Rewrite the expression.

$2 \sqrt{3} + 4 \frac{i}{2}$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Factor $2$ out of $4$ .

$2 \sqrt{3} + 2 (2) \frac{i}{2}$

Step 2.3.2

Cancel the common factor.

$2 \sqrt{3} + 2 \cdot 2 \frac{i}{2}$

Step 2.3.3

Rewrite the expression.

$2 \sqrt{3} + 2 i$