Trigonometry Examples

- \frac{61 π}{6}

$- \frac{61 π}{6}$

Step 1

Find an angle that is positive, less than

2 π

$2 π$ , and coterminal with

- \frac{61 π}{6}

$- \frac{61 π}{6}$ .

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Step 1.1

Add

2 π

$2 π$ to

- \frac{61 π}{6}

$- \frac{61 π}{6}$ until the angle falls between

0

$0$ and

2 π

$2 π$ . In this case,

2 π

$2 π$ needs to be added

6

$6$ times.

- \frac{61 π}{6} + 6 (2 π)

$- \frac{61 π}{6} + 6 (2 π)$

Step 1.2

Simplify.

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Step 1.2.1

Multiply

2

$2$ by

6

$6$ .

- \frac{61 π}{6} + 12 π

$- \frac{61 π}{6} + 12 π$

Step 1.2.2

To write

12 π

$12 π$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

- \frac{61 π}{6} + 12 π \cdot \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

$12 π$ and

\frac{6}{6}

$\frac{6}{6}$ .

- \frac{61 π}{6} + \frac{12 π \cdot 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}

$\frac{- 61 π + 12 π \cdot 6}{6}$

\frac{- 61 π + 12 π \cdot 6}{6}

$\frac{- 61 π + 12 π \cdot 6}{6}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply

6

$6$ by

12

$12$ .

\frac{- 61 π + 72 π}{6}

$\frac{- 61 π + 72 π}{6}$

Step 1.2.4.2

Add

- 61 π

$- 61 π$ and

72 π

$72 π$ .

\frac{11 π}{6}

$\frac{11 π}{6}$

\frac{11 π}{6}

$\frac{11 π}{6}$

\frac{11 π}{6}

$\frac{11 π}{6}$

\frac{11 π}{6}

$\frac{11 π}{6}$

Step 2

Since the angle

\frac{11 π}{6}

$\frac{11 π}{6}$ is in the fourth quadrant, subtract

\frac{11 π}{6}

$\frac{11 π}{6}$ from

2 π

$2 π$ .

2 π - \frac{11 π}{6}

$2 π - \frac{11 π}{6}$

Step 3

Simplify the result.

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Step 3.1

To write

2 π

$2 π$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

2 π \cdot \frac{6}{6} - \frac{11 π}{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

$2 π$ and

\frac{6}{6}

$\frac{6}{6}$ .

\frac{2 π \cdot 6}{6} - \frac{11 π}{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}

$\frac{2 π \cdot 6 - 11 π}{6}$

\frac{2 π \cdot 6 - 11 π}{6}

$\frac{2 π \cdot 6 - 11 π}{6}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

6

$6$ by

2

$2$ .

\frac{12 π - 11 π}{6}

$\frac{12 π - 11 π}{6}$

Step 3.3.2

Subtract

11 π

$11 π$ from

12 π

$12 π$ .

\frac{π}{6}

$\frac{π}{6}$

\frac{π}{6}

$\frac{π}{6}$

\frac{π}{6}

$\frac{π}{6}$