Precalculus Examples

$\frac{19 π}{6}$

Find an angle that is positive, less than $2 π$ , and coterminal with $\frac{19 π}{6}$ .

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Subtract $2 π$ from $\frac{19 π}{6}$ .

$A_{c} = \frac{19 π}{6} - 2 π$

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $\frac{19 π}{6}$ .

$A_{c} = \frac{7 π}{6}$

Since the angle $\frac{7 π}{6}$ is in the third quadrant, the reference angle formula is $A_{r} = A_{c} - π$ .

$A_{r} = \frac{7 π}{6} - π$

Simplify the right side of the equation $A_{r} = \frac{7 π}{6} - π$ .

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To write $- \frac{- π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{7 π}{6} + \frac{- π}{1} \frac{6}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{7 π}{6} + \frac{- π \cdot 6}{1 \cdot 6}$

Multiply $6$ by $1$ .

$\frac{7 π}{6} + \frac{- π \cdot 6}{6}$

Combine the numerators over the common denominator.

$\frac{7 π - π \cdot 6}{6}$

Simplify the numerator.

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Multiply $6$ by $- 1$ .

$\frac{7 π - 6 π}{6}$

Subtract $6 π$ from $7 π$ .

$\frac{π}{6}$

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