Chapter 6.1, Problem 3QR

Solution

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

  $\begin{array}{l}x=5.362\\ y=9.642\end{array}$

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 $7$ .

Let the angle formed in reference triangle be $\theta$ .

In the given figure, the sum of measures of $\theta$ and the straight angle is $220°$ .

Since the measure of straight angle is $180°$ , it follows that:

  $\begin{array}{l}\theta +180°=220°\\ \Rightarrow \theta =40°\end{array}$

Since the angle is formed by the counterclockwise rotation, the angle is $40°$ .

The perpendicular side in reference triangle is $y$ , the adjacent side is $x$ and the hypotenuse is $7$ .

Find $y$ .

  $\begin{array}{l}\sin 40°=\frac{\text{perpendicular}\text{side}}{\text{hypotenuse}}\\ \Rightarrow \sin 40°=\frac{y}{15}\left[\sin 40°\approx 0.6428\right]\\ \Rightarrow y=0.6428\times 15\\ \Rightarrow y=9.642\end{array}$

Find $x$ .

  $\begin{array}{l}\cos 40°=\frac{\text{adjacent}\text{side}}{\text{hypotenuse}}\\ \Rightarrow 0.7660=\frac{x}{7}\left[\cos 40°\approx 0.7660\right]\\ \Rightarrow x=0.7660\times 7=5.362\end{array}$