Algebra Examples

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

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.

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

Step 1.2

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

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 3

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

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Step 3.1

Multiply $- 1$ by $3$ .

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

Step 3.2

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

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

Step 4

Move the negative in front of the fraction.

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