Question

find the reference angle for the given angle.
(a) (2.3pi)
(b) (2.3)
(c) (- 10pi)

Explanation:

Step1: Determine the quadrant for \(2.3\pi\)

Since \(2\pi<2.3\pi < 3\pi\), \(2.3\pi\) is in the third - quadrant. The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the third - quadrant is \(\theta_{r}=\theta - 2\pi\). So, \(\theta_{r}=2.3\pi-2\pi = 0.3\pi\).

Step2: Determine the quadrant for \(2.3\) radians

Since \(\frac{\pi}{2}\approx1.57\) and \(\pi\approx3.14\), and \(1.57 < 2.3<3.14\), the angle \(2.3\) is in the second - quadrant. The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the second - quadrant is \(\theta_{r}=\pi-\theta\). So, \(\theta_{r}=\pi - 2.3\).

Step3: Determine the quadrant for \(- 10\pi\)

Since \(-10\pi\) is a multiple of \(2\pi\) (\(-10\pi=-5\times2\pi\)), the terminal side of the angle \(-10\pi\) lies on the positive \(x\) - axis. The reference angle \(\theta_{r} = 0\).

Answer:

(a) \(0.3\pi\)
(b) \(\pi - 2.3\)
(c) \(0\)