Convert to Rectangular 12(cos((11pi)/6)+isin((11pi)/6))

$12 (\cos (\frac{11 π}{6}) + i \sin (\frac{11 π}{6}))$

Step 1

Simplify each term.

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

$12 (\cos (\frac{π}{6}) + i \sin (\frac{11 π}{6}))$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$12 (\frac{\sqrt{3}}{2} + i \sin (\frac{11 π}{6}))$

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.

$12 (\frac{\sqrt{3}}{2} + i (- \sin (\frac{π}{6})))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$12 (\frac{\sqrt{3}}{2} + i (- \frac{1}{2}))$

Combine $i$ and $\frac{1}{2}$ .

$12 (\frac{\sqrt{3}}{2} - \frac{i}{2})$

Step 2

Simplify terms.

Apply the distributive property.

$12 \frac{\sqrt{3}}{2} + 12 (- \frac{i}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$2 (6) \frac{\sqrt{3}}{2} + 12 (- \frac{i}{2})$

Cancel the common factor.

$2 \cdot 6 \frac{\sqrt{3}}{2} + 12 (- \frac{i}{2})$

Rewrite the expression.

$6 \sqrt{3} + 12 (- \frac{i}{2})$

Cancel the common factor of $2$ .

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

$6 \sqrt{3} + 12 \frac{- i}{2}$

Factor $2$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $6$ .

$6 \sqrt{3} - 6 i$

Convert to Rectangular 12(cos((11pi)/6)+isin((11pi)/6))

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