Question

what is the measure of the reference angle for a -289° angle? a. 89° b. 289° c. 71° d. 19°

Explanation:

Step1: Find coterminal angle

To find the coterminal angle of \(-289^\circ\), we add \(360^\circ\) (since coterminal angles differ by multiples of \(360^\circ\)): \(-289^\circ + 360^\circ = 71^\circ\).

Step2: Determine reference angle

The angle \(71^\circ\) is in the first quadrant. For an angle in the first quadrant, the reference angle is equal to the angle itself. Wait, no—wait, actually, the coterminal angle of \(-289^\circ\) is \(71^\circ\)? Wait, no, let's recalculate. Wait, \(-289^\circ + 360^\circ = 71^\circ\)? Wait, no, \(-289 + 360 = 71\)? Wait, \(360 - 289 = 71\), so \(-289^\circ\) is coterminal with \(71^\circ\)? Wait, no, negative angles are measured clockwise. So \(-289^\circ\) is equivalent to \(360^\circ - 289^\circ = 71^\circ\) when measured counterclockwise? Wait, no, let's think again. The reference angle is the acute angle that the terminal side makes with the x - axis. For a negative angle, we can find its positive coterminal angle by adding \(360^\circ\) until we get a positive angle between \(0^\circ\) and \(360^\circ\). So \(-289^\circ+360^\circ = 71^\circ\). Now, \(71^\circ\) is in the first quadrant (since \(0^\circ<71^\circ < 90^\circ\)? No, \(0^\circ < 71^\circ<90^\circ\)? Wait, \(71^\circ\) is between \(0^\circ\) and \(90^\circ\), so the reference angle for an angle in the first quadrant is the angle itself? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's check the coterminal angle again. \(-289^\circ\) is the same as rotating \(289^\circ\) clockwise. To find the positive coterminal angle, we add \(360^\circ\): \(-289 + 360 = 71^\circ\). Now, \(71^\circ\) is in the first quadrant (since \(0^\circ<71^\circ < 90^\circ\)? No, \(71^\circ\) is between \(0^\circ\) and \(90^\circ\), so the reference angle is equal to the angle, which is \(71^\circ\)? Wait, but let's check the options. Option C is \(71^\circ\). Wait, but let's verify. Wait, maybe I messed up the coterminal angle. Wait, \(-289^\circ\): let's find the positive angle. \(360 - 289 = 71\), so the positive coterminal angle is \(71^\circ\). Since \(71^\circ\) is in the first quadrant, the reference angle is \(71^\circ\)? Wait, but let's think of another way. Suppose we have an angle \(\theta=-289^\circ\). The reference angle is found by: if \(\theta\) is negative, add \(360^\circ\) to get \(\theta'=\theta + 360^\circ=-289 + 360 = 71^\circ\). Now, for \(\theta'\) in the first quadrant (\(0^\circ<\theta'<90^\circ\)? No, \(71^\circ\) is between \(0^\circ\) and \(90^\circ\), so the reference angle is \(\theta' = 71^\circ\). Wait, but let's check the options. Option C is \(71^\circ\). So the reference angle is \(71^\circ\).

Answer:

C. \(71^\circ\)