Question

1. draw the angle with the given measure in standard position. \\( \frac { 17 \pi } { 6 } \\) determine its quadrant and reference angle (in radians). 13. draw the angle with the given measure in standards position. \\( - 654 ^ { \circ } \\). determine its quadrant and reference angle.

Explanation:

Step1: Simplify the angle

For \(\frac{17\pi}{6}\), we can write \(\frac{17\pi}{6}=2\pi+\frac{5\pi}{6}\). Since \(2\pi\) represents a full - rotation, the terminal side of \(\frac{17\pi}{6}\) is the same as the terminal side of \(\frac{5\pi}{6}\).

Step2: Determine the quadrant

The angle \(\frac{5\pi}{6}\) lies in the second quadrant (\(\frac{\pi}{2}<\frac{5\pi}{6}<\pi\)).

Step3: Calculate the reference angle

The formula for the reference angle \(\theta'\) of an angle \(\theta\) in the second quadrant is \(\theta'=\pi - \theta\). For \(\theta=\frac{5\pi}{6}\), the reference angle \(\theta'=\pi-\frac{5\pi}{6}=\frac{\pi}{6}\).

For \(- 654^{\circ}\):

Step1: Find a positive coterminal angle

We add \(360^{\circ}\times2 = 720^{\circ}\) to \(-654^{\circ}\). So, \(-654^{\circ}+720^{\circ}=66^{\circ}\).

Step2: Determine the quadrant

Since \(0^{\circ}<66^{\circ}<90^{\circ}\), the angle \(66^{\circ}\) (and thus \(-654^{\circ}\)) lies in the first quadrant.

Step3: Calculate the reference angle

For an angle \(\theta\) in the first quadrant, the reference angle \(\theta'=\theta\). So the reference angle of \(-654^{\circ}\) (or \(66^{\circ}\)) is \(66^{\circ}\).

Answer:

For \(\frac{17\pi}{6}\): Quadrant - II, Reference angle=\(\frac{\pi}{6}\)
For \(-654^{\circ}\): Quadrant - I, Reference angle = \(66^{\circ}\)