Convert to Rectangular 6(cos((5pi)/3)+isin((5pi)/3))

$6 (\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3}))$

Step 1

Simplify each term.

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

$6 (\cos (\frac{π}{3}) + i \sin (\frac{5 π}{3}))$

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$6 (\frac{1}{2} + i \sin (\frac{5 π}{3}))$

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.

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

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

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

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

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

Step 2

Simplify terms.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $3$ .

$3 - 3 i \sqrt{3}$

Convert to Rectangular 6(cos((5pi)/3)+isin((5pi)/3))

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