Chapter 4.3, Problem 21E

Solution

To choose: The point on the terminal side of the angle $\theta =45°$ .

The point on the terminal side of the angle $\theta =45°$ is $\left(2,2\right)$ .

Given information: The given angle is $\theta =45°$ .

Formula used:

Consider a right triangle with $\theta =45°$ as the angle at the standard position, $x$ as the base of the triangle, $y$ as the perpendicular of the triangle, and $r$ as the distance from the origin to the terminal side of the angle making it the hypotenuse of the triangle.

  $\tan \theta =\frac{y}{x}$

Calculation:

Substitute $45$ for $\theta$ in the tangent formula and calculate the ratio $\frac{y}{x}$ .

  $\begin{array}{c}\tan 45°=\frac{y}{x}\\ 1=\frac{y}{x}\end{array}$

Calculate the ratio of the two coordinates for point a. The point a is $\left(2,2\right)$ .

  $\begin{array}{c}\frac{y}{x}=\frac{2}{2}\\ x=y\end{array}$

Calculate the ratio of the two coordinates for point b. The point b is $\left(1,\sqrt{3}\right)$ .

  $\begin{array}{c}\frac{y}{x}=\frac{\sqrt{3}}{1}\\ \sqrt{3}x=y\end{array}$

Calculate the ratio of the two coordinates for point c. The point c is $\left(\sqrt{3},1\right)$ .

  $\begin{array}{c}\frac{y}{x}=\frac{1}{\sqrt{3}}\\ x=\sqrt{3}y\end{array}$

It can be observed that value of $\frac{y}{x}$ is equal to $1$ for the point a. So, the point a is on the terminal side of $\theta$ .

Therefore, the point on the terminal side of the angle $\theta =45°$ is $\left(2,2\right)$ .