Precalculus Examples

$8 (\cos (330 + i \sin (330)))$

Step 1

To convert degrees to radians, multiply by $\frac{π}{180 °}$ , since a full circle is $360 °$ or $2 π$ radians.

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Factor $4$ out of $8 (\cos (330 + i \sin (330)))$ .

$4 (2 \cos (330 + i \sin (330))) \cdot \frac{π}{180}$ radians

Step 2.1.2

Factor $4$ out of $180$ .

$4 (2 \cos (330 + i \sin (330))) \cdot \frac{π}{4 (45)}$ radians

Step 2.1.3

Cancel the common factor.

$4 (2 \cos (330 + i \sin (330))) \cdot \frac{π}{4 \cdot 45}$ radians

Step 2.1.4

Rewrite the expression.

$2 \cos (330 + i \sin (330)) \cdot \frac{π}{45}$ radians

Step 2.2

Combine $\frac{π}{45}$ and $2$ .

$\frac{π \cdot 2}{45} \cdot \cos (330 + i \sin (330))$ radians

Step 2.3

Combine $\frac{π \cdot 2}{45}$ and $\cos (330 + i \sin (330))$ .

$\frac{π \cdot (2 \cos (330 + i \sin (330)))}{45}$ radians

Step 3

Simplify each term.

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Step 3.1

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.

$\frac{π \cdot (2 \cos (330 + i (- \sin (30))))}{45}$ radians

Step 3.2

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

$\frac{π \cdot (2 \cos (330 + i (- \frac{1}{2})))}{45}$ radians

Step 3.3

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

$\frac{π \cdot (2 \cos (330 - \frac{i}{2}))}{45}$ radians

Step 4

Move $2$ to the left of $π$ .

$\frac{2 π \cos (330 - \frac{i}{2})}{45}$ radians