Question

for each angle below, determine the quadrant in which the angle is located and find the corresponding reference angle. enter the reference angles as reduced fractions involving π.
a. \\( \frac { 10 \pi } { 3 } \\) is found in quadrant and has a reference angle of.
b. \\( \frac { 7 \pi } { 4 } \\) is found in quadrant and has a reference angle of.
c. \\( \frac { 23 \pi } { 6 } \\) is found in quadrant and has a reference angle of.
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Explanation:

Step1: Find the coterminal angle for \(\frac{10\pi}{3}\)

Subtract \(2\pi\) (a full - rotation) from \(\frac{10\pi}{3}\): \(\frac{10\pi}{3}-2\pi=\frac{10\pi - 6\pi}{3}=\frac{4\pi}{3}\). Since \(\pi<\frac{4\pi}{3}<\frac{3\pi}{2}\), it is in Quadrant III. The reference angle for an angle \(\theta\) in Quadrant III is \(\theta-\pi\). So, \(\frac{4\pi}{3}-\pi=\frac{\pi}{3}\).

Step2: Analyze \(\frac{7\pi}{4}\)

Since \(\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi\), it is in Quadrant IV. The reference angle for an angle \(\theta\) in Quadrant IV is \(2\pi-\theta\). So, \(2\pi-\frac{7\pi}{4}=\frac{8\pi - 7\pi}{4}=\frac{\pi}{4}\).

Step3: Find the coterminal angle for \(\frac{23\pi}{6}\)

Subtract \(2\pi\) from \(\frac{23\pi}{6}\): \(\frac{23\pi}{6}-2\pi=\frac{23\pi - 12\pi}{6}=\frac{11\pi}{6}\). Since \(\frac{3\pi}{2}<\frac{11\pi}{6}<2\pi\), it is in Quadrant IV. The reference angle for \(\frac{11\pi}{6}\) (an angle in Quadrant IV) is \(2\pi-\frac{11\pi}{6}=\frac{12\pi - 11\pi}{6}=\frac{\pi}{6}\).

Answer:

a. \(\frac{\pi}{3}\)
b. \(\frac{\pi}{4}\)
c. \(\frac{\pi}{6}\)