Trigonometry Examples

$48 (\cos (\frac{17 π}{6}) + i \sin (\frac{17 π}{6}))$

Step 1

Simplify each term.

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Step 1.1

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

$48 (\cos (\frac{5 π}{6}) + i \sin (\frac{17 π}{6}))$

Step 1.2

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 second quadrant.

$48 (- \cos (\frac{π}{6}) + i \sin (\frac{17 π}{6}))$

Step 1.3

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

$48 (- \frac{\sqrt{3}}{2} + i \sin (\frac{17 π}{6}))$

Step 1.4

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

$48 (- \frac{\sqrt{3}}{2} + i \sin (\frac{5 π}{6}))$

Step 1.5

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

$48 (- \frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

Step 1.6

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

$48 (- \frac{\sqrt{3}}{2} + i \frac{1}{2})$

Step 1.7

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

$48 (- \frac{\sqrt{3}}{2} + \frac{i}{2})$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$48 (- \frac{\sqrt{3}}{2}) + 48 \frac{i}{2}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

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

$48 \frac{- \sqrt{3}}{2} + 48 \frac{i}{2}$

Step 2.2.2

Factor $2$ out of $48$ .

$2 (24) \frac{- \sqrt{3}}{2} + 48 \frac{i}{2}$

Step 2.2.3

Cancel the common factor.

$2 \cdot 24 \frac{- \sqrt{3}}{2} + 48 \frac{i}{2}$

Step 2.2.4

Rewrite the expression.

$24 (- \sqrt{3}) + 48 \frac{i}{2}$

Step 2.3

Multiply $- 1$ by $24$ .

$- 24 \sqrt{3} + 48 \frac{i}{2}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Factor $2$ out of $48$ .

$- 24 \sqrt{3} + 2 (24) \frac{i}{2}$

Step 2.4.2

Cancel the common factor.

$- 24 \sqrt{3} + 2 \cdot 24 \frac{i}{2}$

Step 2.4.3

Rewrite the expression.

$- 24 \sqrt{3} + 24 i$