Activity 4.5.3.
In this activity, we develop the standard properties of the cosecant function, \(q(t) = \csc(t)\text{.}\)
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(a)
Complete the tables below to determine the exact values of the cosecant function at the special points on the unit circle. Enter “\(u\)” for any value at which \(q(t) = \csc(t)\) is undefined.
| \(t\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\frac{3\pi}{4}\) | \(\frac{5\pi}{6}\) | \(\pi\) |
| \(\sin(t)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\csc(t)\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
| \(t\) | \(\frac{7\pi}{6}\) | \(\frac{5\pi}{4}\) | \(\frac{4\pi}{3}\) | \(\frac{3\pi}{2}\) | \(\frac{5\pi}{3}\) | \(\frac{7\pi}{4}\) | \(\frac{11\pi}{6}\) | \(2\pi\) |
| \(\sin(t)\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-1\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) |
| \(\csc(t)\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
(b)
In which quadrants is \(q(t) = \csc(t)\) positive? negative?
(c)
(d)
What is the domain of the cosecant function? What is its range?
(e)
Sketch an accurate, labeled graph of \(q(t) = \csc(t)\) on the figure given at the start of this activity, including the special points that come from the unit circle.
(f)
What is the period of the cosecant function?
