Activity 2.3.2.
Let \(k = g(t)\) be the function that tracks the \(x\)-coordinate of a point traversing the unit circle counterclockwise from \((1,0)\text{.}\) That is, \(g(t) = \cos(t)\text{.}\) Use the information we know about the unit circle that is summarized in Figure 2.3.1 (with 16 labeled special points) to respond to the following questions.
(a)
What is the exact value of \(\cos(\frac{\pi}{6})\text{?}\) of \(\cos(\frac{5\pi}{6})\text{?}\) \(\cos(-\frac{\pi}{3})\text{?}\)
(b)
Complete the following table with the exact values of \(k\) that correspond to the stated inputs.
| \(t\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\frac{3\pi}{4}\) | \(\frac{5\pi}{6}\) | \(\pi\) |
| \(k\) | |||||||||
| \(t\) | \(\pi\) | \(\frac{7\pi}{6}\) | \(\frac{5\pi}{4}\) | \(\frac{4\pi}{3}\) | \(\frac{3\pi}{2}\) | \(\frac{5\pi}{3}\) | \(\frac{7\pi}{4}\) | \(\frac{11\pi}{6}\) | \(2\pi\) |
| \(k\) | |||||||||
(c)
On the axes provided in the following figure, sketch an accurate graph of \(k = \cos(t)\text{.}\) Label the exact location of several key points on the curve.
(d)
What is the exact value of \(\cos( \frac{11\pi}{4} )\text{?}\) of \(\cos( \frac{14\pi}{3} )\text{?}\)
(e)
(f)
How is the graph of \(k = \cos(t)\) different from the graph of \(h = \sin(t)\text{?}\) How are the graphs similar?
