Trigonometry Examples

$- \frac{61 π}{6}$

Step 1

Find an angle that is positive, less than $2 π$ , and coterminal with $- \frac{61 π}{6}$ .

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Step 1.1

Add $2 π$ to $- \frac{61 π}{6}$ until the angle falls between $0$ and $2 π$ . In this case, $2 π$ needs to be added $6$ times.

$- \frac{61 π}{6} + 6 (2 π)$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $2$ by $6$ .

$- \frac{61 π}{6} + 12 π$

Step 1.2.2

To write $12 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{61 π}{6} + 12 π \cdot \frac{6}{6}$

Step 1.2.3

Combine fractions.

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Step 1.2.3.1

Combine $12 π$ and $\frac{6}{6}$ .

$- \frac{61 π}{6} + \frac{12 π \cdot 6}{6}$

Step 1.2.3.2

Combine the numerators over the common denominator.

$\frac{- 61 π + 12 π \cdot 6}{6}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $6$ by $12$ .

$\frac{- 61 π + 72 π}{6}$

Step 1.2.4.2

Add $- 61 π$ and $72 π$ .

$\frac{11 π}{6}$

Step 2

Since the angle $\frac{11 π}{6}$ is in the fourth quadrant, subtract $\frac{11 π}{6}$ from $2 π$ .

$2 π - \frac{11 π}{6}$

Step 3

Simplify the result.

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Step 3.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{11 π}{6}$

Step 3.2

Combine fractions.

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Step 3.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{11 π}{6}$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - 11 π}{6}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $6$ by $2$ .

$\frac{12 π - 11 π}{6}$

Step 3.3.2

Subtract $11 π$ from $12 π$ .

$\frac{π}{6}$