Question

suppose ( a=\frac{7}{6}pi ) radians. ( a ) is an angle in quadrant
i
ii
iii
iv
the reference angle for ( a ) is radians. enter your answer as a reduced fraction involving ( pi ).
question help: video written example message instructor post to forum

Explanation:

Step1: Determine the quadrant

We know that:

• Quadrant I: \(0<\theta <\frac{\pi}{2}\)
• Quadrant II: \(\frac{\pi}{2}<\theta <\pi\)
• Quadrant III: \(\pi<\theta <\frac{3\pi}{2}\)
• Quadrant IV: \(\frac{3\pi}{2}<\theta <2\pi\)

Since \(\pi=\frac{6\pi}{6}\) and \(\frac{3\pi}{2}=\frac{9\pi}{6}\), and \(\frac{6\pi}{6}<\frac{7\pi}{6}<\frac{9\pi}{6}\), so \(A = \frac{7\pi}{6}\) is in Quadrant III.

Step2: Calculate the reference angle

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in Quadrant III is \(\theta_{r}=\theta-\pi\)

Substitute \(\theta=\frac{7\pi}{6}\) into the formula: \(\theta_{r}=\frac{7\pi}{6}-\pi=\frac{7\pi}{6}-\frac{6\pi}{6}=\frac{\pi}{6}\)

Answer:

III, \(\frac{\pi}{6}\)