Question

question 3
12 pts
choose all angles that terminate in quadrant 2.
(there is more than one answer)
5 radians
-100 degrees
170 degrees
-225 degrees
290 degrees
\\(\frac{9\pi}{7}\\)
\\(-\frac{7\pi}{6}\\)

Explanation:

Step1: Recall quadrant - 2 angle range

Angles in quadrant 2 in degrees range from 90° to 180°. In radians, since \(180^{\circ}=\pi\) radians and \(90^{\circ}=\frac{\pi}{2}\) radians, the range is \(\frac{\pi}{2}<\theta<\pi\) radians for positive - angles. For negative - angles, we add 360° or \(2\pi\) radians to find the equivalent positive angle.

Step2: Analyze each option

• For 5 radians: Since \(\pi\approx3.14\) and \(2\pi\approx6.28\), \(5\) radians is in the third quadrant (\(\pi < 5<\frac{3\pi}{2}\)).
• For \(- 100^{\circ}\): Add 360° to get \(360 - 100=260^{\circ}\), which is in the third quadrant.
• For \(170^{\circ}\): Since \(90^{\circ}<170^{\circ}<180^{\circ}\), it is in quadrant 2.
• For \(-225^{\circ}\): Add 360° to get \(360 - 225 = 135^{\circ}\), which is in quadrant 2.
• For \(290^{\circ}\): It is in the fourth quadrant (\(270^{\circ}<290^{\circ}<360^{\circ}\)).
• For \(\frac{9\pi}{7}\): Since \(\pi\approx3.14\), \(\frac{9\pi}{7}\approx\frac{9\times3.14}{7}\approx4.03\), which is in the third quadrant (\(\pi<\frac{9\pi}{7}<\frac{3\pi}{2}\)).
• For \(-\frac{7\pi}{6}\): Add \(2\pi\) to get \(2\pi-\frac{7\pi}{6}=\frac{12\pi - 7\pi}{6}=\frac{5\pi}{6}\). Since \(\frac{\pi}{2}<\frac{5\pi}{6}<\pi\), it is in quadrant 2.

Answer:

170 degrees, - 225 degrees, \(-\frac{7\pi}{6}\)