Trigonometry Examples

$6 (\cos (300) + i \sin (300))$

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.

$6 (\cos (60) + i \sin (300))$

Step 1.2

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

$6 (\frac{1}{2} + i \sin (300))$

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.

$6 (\frac{1}{2} + i (- \sin (60)))$

Step 1.4

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

$6 (\frac{1}{2} + i (- \frac{\sqrt{3}}{2}))$

Step 1.5

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

$6 (\frac{1}{2} - \frac{i \sqrt{3}}{2})$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$6 (\frac{1}{2}) + 6 (- \frac{i \sqrt{3}}{2})$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{1}{2} + 6 (- \frac{i \sqrt{3}}{2})$

Step 2.2.2

Cancel the common factor.

$2 \cdot 3 \frac{1}{2} + 6 (- \frac{i \sqrt{3}}{2})$

Step 2.2.3

Rewrite the expression.

$3 + 6 (- \frac{i \sqrt{3}}{2})$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

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

$3 + 6 \frac{- i \sqrt{3}}{2}$

Step 2.3.2

Factor $2$ out of $6$ .

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

Step 2.3.3

Cancel the common factor.

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

Step 2.3.4

Rewrite the expression.

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

Step 2.4

Multiply $- 1$ by $3$ .

$3 - 3 i \sqrt{3}$