Chapter 6.1, Problem 1QR

Solution

To find: The value of $x$ and $y$ .

  $\begin{array}{l}y=4.5\\ x=7.794\end{array}$

Given information:

Find the value of $x$ and $y$ .

  

Concept used:

The angle is formed by the counterclockwise rotation. So, the angle is positive.

Use the trigonometric function of angle to find the value of $x$ and $y$ .

Explanation:

The distance of $\left(x,y\right)$ from the origin is $9$ .

Since the angle is formed by the counterclockwise rotation, the angle is $30°$ .

The perpendicular side in reference triangle is $y$ , the adjacent side is $x$ and the hypotenuse is

  $9$ .

Find $y$ .

  $\begin{array}{l}\sin 30°=\frac{\text{perpendicular}\text{side}}{\text{hypotenuse}}\\ \Rightarrow \frac{1}{2}=\frac{y}{9}\left[\sin 30°=\frac{1}{2}\right]\\ \Rightarrow y=\frac{9}{2}=4.5\end{array}$

Find $x$ .

  $\begin{array}{l}\cos 30°=\frac{\text{adjacent}\text{side}}{\text{hypotenuse}}\\ \Rightarrow \frac{\sqrt{3}}{2}=\frac{x}{9}\left[\cos 30°=\frac{\sqrt{3}}{2}\right]\\ \Rightarrow x=\frac{9\sqrt{3}}{2}\\ \Rightarrow x=\frac{9\times 1.732}{2}\left[\text{Take}\sqrt{3}=1.732\right]\\ \Rightarrow x=7.794\end{array}$