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].
