Question

Find$t$and the terminal point determined by $t$for each point in the figure. In Exercise 22, $t$increase in increments of $\frac{\pi }{6}$.

Short answer

The value of $t$ is $0,\frac{2\pi }{6},\frac{3\pi }{6},\frac{2\pi }{3},\frac{5\pi }{6},\pi ,\frac{7\pi }{6},\frac{4\pi }{3},\frac{3\pi }{2},\frac{5\pi }{3},\frac{11\pi }{6},2\pi$, and the terminal points are $\begin{array}{l}\left(0,1\right),\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right),\left(1,0\right),\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right),\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right),\left(0,-1\right),\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right),\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\\ ,\left(-1,0\right),\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right),\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right),\left(0,1\right)\end{array}$.

Step-by-step solution

Given information.

The given figure is,

Calculation.

Since the value of $t$ is increaseing with $\frac{\pi }{6}$. So, make the table and calculate the value of each point by using the terminal point $\left(\sin t,\cos t\right)$.

Calculate the value of terminal point on each value of $t$.

$\begin{array}{ccc}t& \left(\sin t,\cos t\right)& \left(x,y\right)\\ 0& \left(\sin \left(0\right),\cos \left(0\right)\right)& \left(0,1\right)\\ \frac{2\pi }{6}& \left(\sin \left(\frac{2\pi }{6}\right),\cos \left(\frac{2\pi }{6}\right)\right)& \left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\\ \frac{3\pi }{6}& \left(\sin \left(\frac{3\pi }{6}\right),\cos \left(\frac{3\pi }{6}\right)\right)& \left(1,0\right)\\ \frac{2\pi }{3}& \left(\sin \left(\frac{2\pi }{3}\right),\cos \left(\frac{2\pi }{3}\right)\right)& \left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\\ \frac{5\pi }{6}& \left(\sin \left(\frac{5\pi }{6}\right),\cos \left(\frac{5\pi }{6}\right)\right)& \left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\\ \pi & \left(\sin \left(\pi \right),\cos \left(\pi \right)\right)& \left(0,-1\right)\\ \frac{7\pi }{6}& \left(\sin \left(\frac{7\pi }{6}\right),\cos \left(\frac{7\pi }{6}\right)\right)& \left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\\ \frac{4\pi }{3}& \left(\sin \left(\frac{4\pi }{3}\right),\cos \left(\frac{4\pi }{3}\right)\right)& \left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\\ \frac{3\pi }{2}& \left(\sin \left(\frac{3\pi }{2}\right),\cos \left(\frac{3\pi }{2}\right)\right)& \left(-1,0\right)\\ \frac{5\pi }{3}& \left(\sin \left(\frac{5\pi }{3}\right),\cos \left(\frac{5\pi }{3}\right)\right)& \left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)\\ \frac{11\pi }{6}& \left(\sin \left(\frac{11\pi }{6}\right),\cos \left(\frac{11\pi }{6}\right)\right)& \left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)\\ 2\pi & \left(\sin \left(2\pi \right),\cos \left(2\pi \right)\right)& \left(0,1\right)\end{array}$