Question

find the reference number for each value of t. (a) ( t=\frac{4 pi}{3} ) (b) ( t=\frac{5 pi}{3} ) (c) ( t=-\frac{7 pi}{6} ) (d) ( t = 3.7 )

Explanation:

Step1: Determine the quadrant for \(t = \frac{4\pi}{3}\)

Since \(\pi<\frac{4\pi}{3}<\frac{3\pi}{2}\), \(t=\frac{4\pi}{3}\) is in the third - quadrant.
The formula for the reference number \(\overline{t}\) in the third - quadrant is \(\overline{t}=t - \pi\).
So, \(\overline{t}=\frac{4\pi}{3}-\pi=\frac{4\pi - 3\pi}{3}=\frac{\pi}{3}\)

Step2: Determine the quadrant for \(t=\frac{5\pi}{3}\)

Since \(\frac{3\pi}{2}<\frac{5\pi}{3}<2\pi\), \(t = \frac{5\pi}{3}\) is in the fourth - quadrant.
The formula for the reference number \(\overline{t}\) in the fourth - quadrant is \(\overline{t}=2\pi - t\).
So, \(\overline{t}=2\pi-\frac{5\pi}{3}=\frac{6\pi - 5\pi}{3}=\frac{\pi}{3}\)

Step3: Determine the quadrant for \(t =-\frac{7\pi}{6}\)

First, find the positive coterminal angle. Add \(2\pi\) to \(t\): \(t'=-\frac{7\pi}{6}+2\pi=\frac{-7\pi + 12\pi}{6}=\frac{5\pi}{6}\)
Since \(\frac{\pi}{2}<\frac{5\pi}{6}<\pi\), the angle is in the second - quadrant.
The formula for the reference number \(\overline{t}\) in the second - quadrant is \(\overline{t}=\pi - t\) (for the positive coterminal angle \(t'=\frac{5\pi}{6}\)). So, \(\overline{t}=\pi-\frac{5\pi}{6}=\frac{\pi}{6}\)

Step4: Determine the quadrant for \(t = 3.7\)

Since \(\pi\approx3.14\) and \(2\pi\approx6.28\), and \(3.14<3.7<6.28\).
The formula for the reference number \(\overline{t}\) when \(t\) is in the third or fourth quadrant (here \(t = 3.7\) is in the third quadrant as \(3.7-\pi\approx3.7 - 3.14=0.56\) and \(3.7<\pi + 1.57\approx4.71\) is wrong, actually \(t = 3.7\) is in the third quadrant). The formula for the reference number \(\overline{t}\) is \(\overline{t}=t-\pi\) (if \(t\) is in the third quadrant, \(t\in(\pi,\frac{3\pi}{2})\)). \(\overline{t}=3.7-\pi\approx3.7 - 3.14 = 0.56\)

Answer:

(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{3}\)
(c) \(\frac{\pi}{6}\)
(d) \(3.7-\pi\approx0.56\)