Select all the correct answers.

If the measure of angle [tex]\theta[/tex] is [tex]\frac{2 \pi}{3}[/tex], which statements are true?

A. The measure of the reference angle is [tex]45^{\circ}[/tex].
B. [tex]\tan(\theta) = -\sqrt{3}[/tex]
C. [tex]\cos(\theta) = \frac{\sqrt{3}}{2}[/tex]
D. The measure of the reference angle is [tex]30^{\circ}[/tex].
E. The measure of the reference angle is [tex]60^{\circ}[/tex].
F. [tex]\sin(\theta) = -\frac{1}{2}[/tex]



Answer :

Let's solve the problem step-by-step to determine which statements are true for the given angle [tex]\(\theta = \frac{2\pi}{3}\)[/tex].

First, let's recall the unit circle and the relevant trigonometric values:

1. Calculate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[\tan\left(\frac{2\pi}{3}\right) = -1.7320508075688783\][/tex]
So, [tex]\(\tan(\theta) \neq -\sqrt{3}\)[/tex].

2. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[\cos\left(\frac{2\pi}{3}\right) = -0.4999999999999998\][/tex]
[tex]\(\cos(\theta) \neq \frac{\sqrt{3}}{2}\)[/tex].

3. Calculate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[\sin\left(\frac{2\pi}{3}\right) = 0.8660254037844387\][/tex]
[tex]\(\sin(\theta) \neq -\frac{1}{2}\)[/tex].

4. Determine the reference angle:
The reference angle for [tex]\(\theta = \frac{2\pi}{3}\)[/tex] is calculated as:
[tex]\[ \text{Reference angle} = \pi - \frac{2\pi}{3} = \frac{\pi}{3} \][/tex]

Convert this reference angle to degrees:
[tex]\[\text{Reference angle in degrees} = \frac{\pi}{3} \times \frac{180}{\pi} = 60^{\circ}\][/tex]
So, the reference angle is [tex]\(60^{\circ}\)[/tex].

Now, let's evaluate the given statements:

- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]:
This statement is false because the reference angle is [tex]\(60^{\circ}\)[/tex].

- [tex]\(\tan(\theta) = -\sqrt{3}\)[/tex]:
This statement is false because [tex]\(\tan(\theta) = -1.7320508075688783\)[/tex].

- [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]:
This statement is false because [tex]\(\cos(\theta) = -0.4999999999999998\)[/tex].

- The measure of the reference angle is [tex]\(30^{\circ}\)[/tex]:
This statement is false because the reference angle is [tex]\(60^{\circ}\)[/tex].

- The measure of the reference angle is [tex]\(60^{\circ}\)[/tex]:
This statement is true because the reference angle is [tex]\(60^{\circ}\)[/tex].

- [tex]\(\sin(\theta) = -\frac{1}{2}\)[/tex]:
This statement is false because [tex]\(\sin(\theta) = 0.8660254037844387\)[/tex].

Therefore, the only correct statement is:
- The measure of the reference angle is [tex]\(60^{\circ}\)[/tex].