Precalculus Examples

$\sqrt{7} (\cos (- 120) + i \sin (- 120))$

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 third quadrant.

$\sqrt{7} (- \cos (60) + i \sin (- 120))$

Step 1.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\sqrt{7} (- \frac{1}{2} + i \sin (- 120))$

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 third quadrant.

$\sqrt{7} (- \frac{1}{2} + i (- \sin (60)))$

Step 1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{7} (- \frac{1}{2} + i (- \frac{\sqrt{3}}{2}))$

Step 1.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$\sqrt{7} (- \frac{1}{2} - \frac{i \sqrt{3}}{2})$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$\sqrt{7} (- \frac{1}{2}) + \sqrt{7} (- \frac{i \sqrt{3}}{2})$

Step 2.2

Combine $\sqrt{7}$ and $\frac{1}{2}$ .

$- \frac{\sqrt{7}}{2} + \sqrt{7} (- \frac{i \sqrt{3}}{2})$

Step 3

Multiply $\sqrt{7} (- \frac{i \sqrt{3}}{2})$ .

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Step 3.1

Combine $\sqrt{7}$ and $\frac{i \sqrt{3}}{2}$ .

$- \frac{\sqrt{7}}{2} - \frac{\sqrt{7} (i \sqrt{3})}{2}$

Step 3.2

Combine using the product rule for radicals.

$- \frac{\sqrt{7}}{2} - \frac{\sqrt{3 \cdot 7} i}{2}$

Step 3.3

Multiply $3$ by $7$ .

$- \frac{\sqrt{7}}{2} - \frac{\sqrt{21} i}{2}$