Question

consider the following.
( t = \frac { 7 pi } { 3 } )
(a) find the reference number ( overline { t } ) for the value of ( t ).
( overline { t } = )
(b) find the terminal point determined by ( t ).
( ( x , y ) = ( quad ) )

Explanation:

Step1: Find the reference number

The formula for the reference number \(\overline{t}\) when \(t=\frac{7\pi}{3}\) is \(\overline{t}=t - 2\pi n\), where \(n\) is the largest integer such that \(2\pi n\leq t\).
Since \(2\pi=\frac{6\pi}{3}\), and \(\frac{7\pi}{3}- 2\pi=\frac{7\pi}{3}-\frac{6\pi}{3}=\frac{\pi}{3}\), so \(\overline{t}=\frac{\pi}{3}\).

Step2: Find the terminal point

The terminal point \((x,y)\) for a given \(t\) is based on the unit - circle definitions \(x = \cos t\) and \(y=\sin t\).
Since \(t=\frac{7\pi}{3}\), and \(\cos\frac{7\pi}{3}=\cos(2\pi+\frac{\pi}{3})=\cos\frac{\pi}{3}=\frac{1}{2}\), \(\sin\frac{7\pi}{3}=\sin(2\pi + \frac{\pi}{3})=\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\)

Answer:

(a) \(\overline{t}=\frac{\pi}{3}\)
(b) \((x,y)=(\frac{1}{2},\frac{\sqrt{3}}{2})\)