Trigonometry Examples

$8 (\cos (150) + i \sin (150))$

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. Make the expression negative because cosine is negative in the second quadrant.

$8 (- \cos (30) + i \sin (150))$

Step 1.2

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

$8 (- \frac{\sqrt{3}}{2} + i \sin (150))$

Step 1.3

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

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

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

Step 2.2.2

Factor $2$ out of $8$ .

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

Step 2.2.3

Cancel the common factor.

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

Step 2.2.4

Rewrite the expression.

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

Step 2.3

Multiply $- 1$ by $4$ .

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Factor $2$ out of $8$ .

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

Step 2.4.2

Cancel the common factor.

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

Step 2.4.3

Rewrite the expression.

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