Trigonometry Examples

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

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.

$8 (\cos (30) + i \sin (330))$

Step 1.2

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

$8 (\frac{\sqrt{3}}{2} + i \sin (330))$

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.

$8 (\frac{\sqrt{3}}{2} + i (- \sin (30)))$

Step 1.4

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

$8 (\frac{\sqrt{3}}{2} + i (- \frac{1}{2}))$

Step 1.5

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

$8 (\frac{\sqrt{3}}{2} - \frac{i}{2})$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$8 \frac{\sqrt{3}}{2} + 8 (- \frac{i}{2})$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{\sqrt{3}}{2} + 8 (- \frac{i}{2})$

Step 2.2.2

Cancel the common factor.

$2 \cdot 4 \frac{\sqrt{3}}{2} + 8 (- \frac{i}{2})$

Step 2.2.3

Rewrite the expression.

$4 \sqrt{3} + 8 (- \frac{i}{2})$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Move the leading negative in $- \frac{i}{2}$ into the numerator.

$4 \sqrt{3} + 8 \frac{- i}{2}$

Step 2.3.2

Factor $2$ out of $8$ .

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

Step 2.3.3

Cancel the common factor.

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

Step 2.3.4

Rewrite the expression.

$4 \sqrt{3} + 4 (- i)$

Step 2.4

Multiply $- 1$ by $4$ .

$4 \sqrt{3} - 4 i$