Question

Find $t$and the terminal point determined by $t$for each point in the figure. In Exercise 21, $t$increase in increments of $\frac{\pi }{4}$.

Short answer

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

Step-by-step solution

Step 1. Given information.

The given figure is,

Step 2. Calculation.

Since the value of $t$ is increaseing with $\frac{\pi }{4}$. 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 $t$.

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