Calculus Examples

$2 (\cos (120 °) + i \sin (120 °))$

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 second quadrant.

$2 (- \cos (60) + i \sin (120 °))$

Step 1.2

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

$2 (- \frac{1}{2} + i \sin (120 °))$

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

$- 1 + i \sqrt{3}$