Question

numerical response: (1 mark each)

1. if the point ( pleft(-\frac{sqrt{3}}{2}, \frac{1}{2}

ight) ) lies on the terminal arm of a rotation angle where ( 0^{circ} leq \theta leq 360^{circ} ), then themeasure of ( \theta ) , to the nearest degree is: ____________
use the information below to answer the next question
the following statements are made about the unit circle, where ( \theta ) is an angle drawn instandard position on the coordinate plane.
statement 1 the point ( aleft(\frac{sqrt{5}}{3}, \frac{-2}{3}
ight) ) lies on the unit circle.
statement 2 the point ( b(0.75,0.25) ) lies on the unit circle.
statement 3 for any point ( (x, y) ) on the unit circle, ( \tan \theta=\frac{y}{x} ).
statement 4 any point that lies on the unit circle can be described as ( (sin \theta, cos \theta) ).

Explanation:

Step1: Use the formula for \(\tan\theta\)

For a point \((x,y)\) on the terminal arm of an angle \(\theta\) in standard position, \(\tan\theta=\frac{y}{x}\). Given \(x =-\frac{\sqrt{3}}{2}\) and \(y=\frac{1}{2}\), then \(\tan\theta=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}\approx - 0.577\).

Step2: Determine the reference angle

We know that \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\approx0.577\). Since \(x<0\) and \(y > 0\), the angle \(\theta\) is in the second quadrant.

Step3: Calculate the angle \(\theta\)

The formula for an angle \(\theta\) in the second quadrant is \(\theta = 180^{\circ}-\text{reference angle}\). The reference angle is \(30^{\circ}\), so \(\theta=180^{\circ}- 30^{\circ}=150^{\circ}\)

Answer:

\(150\)