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Coterminal Angles

Coterminal angles are angles in standard position (angles with the initial side on the positive $x$ -axis) that have a common terminal side. For example $30 °$ , $- 330 °$ and $390 °$ are all coterminal.

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To find a positive and a negative angle coterminal with a given angle, you can add and subtract $360 °$ if the angle is measured in degrees or $2 π$ if the angle is measured in radians .

Example 1:

Find a positive and a negative angle coterminal with a $55 °$ angle.

$\begin{array}{l} 55 ° - 360 ° = - 305 ° \\ 55 ° + 360 ° = 415 ° \end{array}$

A $- 305 °$ angle and a $415 °$ angle are coterminal with a $55 °$ angle.

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Example 2:

Find a positive and a negative angle coterminal with a $\frac{π}{3}$ angle.

$\begin{array}{l} \frac{π}{3} + 2 π = \frac{7 π}{3} \\ \frac{π}{3} - 2 π = - \frac{5 π}{3} \end{array}$

A $\frac{7 π}{3}$ angle and a $- \frac{5 π}{3}$ angle are coterminal with a $\frac{π}{3}$ angle.

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