Convert to Rectangular 4(cos((4pi)/3)+isin((4pi)/3))

$4 (\cos (\frac{4 π}{3}) + i \sin (\frac{4 π}{3}))$

Step 1

Simplify each term.

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.

$4 (- \cos (\frac{π}{3}) + i \sin (\frac{4 π}{3}))$

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

$4 (- \frac{1}{2} + i \sin (\frac{4 π}{3}))$

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 third quadrant.

$4 (- \frac{1}{2} + i (- \sin (\frac{π}{3})))$

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

$4 (- \frac{1}{2} + i (- \frac{\sqrt{3}}{2}))$

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

$4 (- \frac{1}{2} - \frac{i \sqrt{3}}{2})$

Step 2

Simplify terms.

Apply the distributive property.

$4 (- \frac{1}{2}) + 4 (- \frac{i \sqrt{3}}{2})$

Cancel the common factor of $2$ .

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

$4 (\frac{- 1}{2}) + 4 (- \frac{i \sqrt{3}}{2})$

Factor $2$ out of $4$ .

$2 (2) \frac{- 1}{2} + 4 (- \frac{i \sqrt{3}}{2})$

Cancel the common factor.

$2 \cdot 2 \frac{- 1}{2} + 4 (- \frac{i \sqrt{3}}{2})$

Rewrite the expression.

$2 \cdot - 1 + 4 (- \frac{i \sqrt{3}}{2})$

Multiply $2$ by $- 1$ .

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

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $2$ .

$- 2 - 2 i \sqrt{3}$

Convert to Rectangular 4(cos((4pi)/3)+isin((4pi)/3))

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