Chapter 13.2, Problem 62PPSE

Solution

To find: The coordinates of the point at which the terminal side of angle $120°$ intersects the unit circle.

The coordinates of the terminal side of angle is $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ . So, option $\left(D\right)$ is correct.

Given:

The measure of angle $\left(\theta \right)=\frac{2\pi }{3}$ .

Concept used:

Use the unit circle or radius $1$ and construct a $30°-60°-90°$ triangle where the sides opposite to angle $30°$ is $\left(a\right)$ which is the shorter leg, side opposite to angle $60°$ is $\sqrt{3}a$ and hypotenuse of triangle is $2a$ as shown below.

  

Calculation:

Draw a unit circle of radius $1$ with vertex at $\left(0,0\right)$ where hypotenuse makes an angle of $\left(\theta \right)=120°$ with positive direction of $x-$ axis. Therefore, by linear pair of angles the other angle will be $60°$ which results a $30°-60°-90°$ triangle as shown below.

  

Now using the concept of $30°-60°-90°$ triangle as stated above, the side length opposite to angle $30°$ is $-\frac{1}{2}$ and side opposite to angle $60°$ is $\frac{\sqrt{3}}{2}$ because the hypotenuse of unit circle triangle is $1$ . Therefore, the coordinates of point where the terminal side of angle $120°$ intersects the unit circle is $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ .

Conclusion:

The coordinates of the terminal side of angle is $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ . So, option $\left(D\right)$ is correct.