Activity 4.5.4.
In this activity, we develop the standard properties of the cotangent function, \(r(t) = \cot(t)\text{.}\)
(a)
Complete the following tables to determine the exact values of the cotangent function at the special points on the unit circle. Enter “u” for any value at which \(r(t) = \cot(t)\) is undefined.
| \(t\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\frac{3\pi}{4}\) | \(\frac{5\pi}{6}\) | \(\pi\) |
| \(\sin(t)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\cos(t)\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-1\) |
| \(\tan(t)\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\frac{3}{\sqrt{3}}\) | u | \(-\frac{3}{\sqrt{3}}\) | \(-1\) | \(-\frac{1}{\sqrt{3}}\) | \(0\) |
| \(\cot(t)\) |
| \(t\) | \(\frac{7\pi}{6}\) | \(\frac{5\pi}{4}\) | \(\frac{4\pi}{3}\) | \(\frac{3\pi}{2}\) | \(\frac{5\pi}{3}\) | \(\frac{7\pi}{4}\) | \(\frac{11\pi}{6}\) | \(2\pi\) |
| \(\sin(t)\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-1\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) |
| \(\cos(t)\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\tan(t)\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\frac{3}{\sqrt{3}}\) | u | \(-\frac{3}{\sqrt{3}}\) | \(-1\) | \(-\frac{1}{\sqrt{3}}\) | \(0\) |
| \(\cot(t)\) |
(b)
In which quadrants is \(r(t) = \cot(t)\) positive? negative?
(c)
(d)
What is the domain of the cotangent function? What is its range?
(e)
Sketch an accurate, labeled graph of \(r(t) = \cot(t)\) on the axes provided below, including the special points that come from the unit circle.
ADD ALT TEXT TO THIS IMAGE
(f)
On intervals where the function is defined at every point in the interval, is \(r(t) = \cot(t)\) always increasing, always decreasing, or neither?
(g)
What is the period of the cotangent function?
(h)
How would you describe the relationship between the graphs of the tangent and cotangent functions?
