Question

find the reference number for each value of t.
(a) ( t=\frac{5 pi}{7} )
( \frac{2 pi}{7} )
(b) ( t=-\frac{7 pi}{9} )
( \frac{2 pi}{9} )
(c) ( t=-3 )
(d) ( t=5 )

Explanation:

Step1: Determine the quadrant for \(t = - 3\)

Since \(\pi\approx3.14\), then \(-3\) is equivalent to \(2\pi - 3\approx6.28 - 3 = 3.28\). And \(\pi<3.28<\frac{3\pi}{2}\), so \(t=-3\) is in the third - quadrant.
The formula for the reference number \(\overline{t}\) when \(t\) is in the third - quadrant is \(\overline{t}=t-\pi\).
\(\overline{t}=2\pi - 3-\pi=\pi - 3\approx3.14 - 3=0.14\)

Step2: Determine the quadrant for \(t = 5\)

Since \(2\pi\approx6.28\), then \(5\) is in the fourth - quadrant (\(\frac{3\pi}{2}\approx4.71<5<2\pi\)).
The formula for the reference number \(\overline{t}\) when \(t\) is in the fourth - quadrant is \(\overline{t}=2\pi - t\)
\(\overline{t}=2\pi - 5\approx6.28 - 5 = 1.28\)

Answer:

(c) \(\pi - 3\)
(d) \(2\pi - 5\)