Precalculus Examples

$\frac{11 π}{3}$

Find an angle that is positive, less than $2 π$ , and coterminal with $\frac{11 π}{3}$ .

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Subtract $2 π$ from $\frac{11 π}{3}$ .

$A_{c} = \frac{11 π}{3} - 2 π$

The resulting angle of $\frac{5 π}{3}$ is positive, less than $2 π$ , and coterminal with $\frac{11 π}{3}$ .

$A_{c} = \frac{5 π}{3}$

Since the angle $\frac{5 π}{3}$ is in the fourth quadrant, the reference angle formula is $A_{r} = 2 π - A_{c}$ .

$A_{r} = 2 π - \frac{5 π}{3}$

Simplify the right side of the equation $A_{r} = 2 π - \frac{5 π}{3}$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 π}{1} \frac{3}{3} - \frac{5 π}{3}$

Write each expression with a common denominator of $3$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 3}{1 \cdot 3} - \frac{5 π}{3}$

Multiply $3$ by $1$ .

$\frac{2 π \cdot 3}{3} - \frac{5 π}{3}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - (5 π)}{3}$

Simplify the numerator.

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Multiply $3$ by $2$ .

$\frac{6 π - (5 π)}{3}$

Multiply $5$ by $- 1$ .

$\frac{6 π - 5 π}{3}$

Subtract $5 π$ from $6 π$ .

$\frac{π}{3}$

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