Question

for an angle of the following measure, state in which quadrant the terminal side lies.

• 175°

the terminal side of -175° lies in quadrant
(type i, ii, iii, or iv.)

Explanation:

Step1: Recall negative angle rotation

A negative angle is measured clockwise from the positive x - axis. So, we can find the positive coterminal angle of \(- 175^{\circ}\) by adding \(360^{\circ}\) to it. The formula for finding a positive coterminal angle \(\theta_{cot}\) of an angle \(\theta\) is \(\theta_{cot}=\theta + 360^{\circ}n\) (where \(n\) is an integer, here \(n = 1\) to get a positive angle). So, \(\theta=-175^{\circ}\), then \(\theta_{cot}=-175^{\circ}+360^{\circ}=185^{\circ}\).

Step2: Determine quadrant based on angle range

• Quadrant I: \(0^{\circ}<\theta < 90^{\circ}\)
• Quadrant II: \(90^{\circ}<\theta < 180^{\circ}\)
• Quadrant III: \(180^{\circ}<\theta < 270^{\circ}\)
• Quadrant IV: \(270^{\circ}<\theta < 360^{\circ}\)

Since \(180^{\circ}<185^{\circ}<270^{\circ}\), the angle \(185^{\circ}\) (and thus \(-175^{\circ}\) as they are coterminal) lies in Quadrant III.

Answer:

III