Question

can a reference angle have a negative measure? justify your reasoning.
o a. yes. any angle in standard position with a terminal side in quadrants iii or iv has a negative reference angle.
o b. no. a reference angle must be an acute angle, which by definition must measure between 0 and 90.
o c. no. a reference angle must be on the unit circle, which by definition must measure between 0 and 360.
o d. yes. the reference angle can be negative if the angle is positioned with the terminal side on the positive x - axis rather than in standard position.
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Explanation:

A reference angle is defined as the acute (between \(0^{\circ}\) and \(90^{\circ}\)) angle that the terminal side of an angle in standard position makes with the \(x -\)axis. By the definition of an acute angle (which is part of the reference - angle definition), it cannot be negative.

• Option A: Incorrect. Angles in Quadrants III or IV have positive reference angles. For example, if the angle \(\theta = 210^{\circ}\) (Quadrant III), the reference angle \(\theta_{r}=210^{\circ}- 180^{\circ}=30^{\circ}\). If \(\theta = 300^{\circ}\) (Quadrant IV), \(\theta_{r}=360^{\circ}-300^{\circ} = 60^{\circ}\).
• Option B: Correct. Since a reference angle is an acute angle (by definition, \(0<\theta_{r}<90\) when measured in degrees), it cannot be negative.
• Option C: Incorrect. The unit - circle definition is not the reason why the reference angle cannot be negative. The key is that it is an acute angle.
• Option D: Incorrect. Angles in standard position (including those with terminal sides on the positive \(x -\)axis, which have a reference angle of \(0^{\circ}\)) do not have negative reference angles. The reference - angle concept is related to the acute - angle formed with the \(x -\)axis, not the position of the terminal side (as long as it's in standard position).

Answer:

B. No. A reference angle must be an acute angle, which by definition must measure between 0 and 90.