Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 1.5

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.

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

Step 1.6

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

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

Step 1.7

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

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.

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

Rewrite the expression.

$1 - i \sqrt{3}$