Find coterminal angles \(A_c\) for a given angle \(A\).
Angles are called coterminal if they share the same terminal side when drawn in standard position. For example, angles \(\alpha = 30^\circ\) and \(\beta = -330^\circ\) are coterminal.
Coterminal angles \(A_c\) can be found by adding or subtracting integer multiples of a full rotation:
If \(A\) is in degrees: \[ A_c = A + k \times 360^\circ \] If \(A\) is in radians: \[ A_c = A + k \times 2\pi \] where \(k\) is any positive or negative integer.
Find a positive and a negative coterminal angle for \(A = -200^\circ\).
Solution:
To find a positive coterminal angle, add 360° (or multiples of 360°) until the result is positive: \[ A_c = -200^\circ + 360^\circ = 160^\circ \] To find a negative coterminal angle, subtract 360° (or multiples of 360°): \[ A_c = -200^\circ - 360^\circ = -560^\circ \] Thus, one positive coterminal angle is \(160^\circ\) and one negative coterminal angle is \(-560^\circ\).
Find a coterminal angle \(A_c\) for \(A = -\frac{17\pi}{3}\) such that \(0 \le A_c < 2\pi\).
Solution:
We repeatedly add \(2\pi\) until the angle is in the range \([0, 2\pi)\): \[ A = -\frac{17\pi}{3} = -2\cdot 2\pi - \frac{5\pi}{3} \] To make it positive, add \(3\cdot 2\pi = 6\pi\): \[ A_c = -\frac{17\pi}{3} + 6\pi = \frac{\pi}{3} \] Hence, the coterminal angle in the desired range is \(\frac{\pi}{3}\).
Find a coterminal angle \(A_c\) for \(A = \frac{35\pi}{4}\) such that \(0 \le A_c < 2\pi\).
Solution:
Express \(A\) in terms of multiples of \(2\pi\): \[ \frac{35\pi}{4} = 4\cdot 2\pi + \frac{3\pi}{4} \] Subtract \(4\cdot 2\pi\) to get an angle less than \(2\pi\): \[ A_c = \frac{35\pi}{4} - 8\pi = \frac{3\pi}{4} \]
Solve the following and check your solutions below:
Solutions: