To address this problem, we need to verify various statements regarding the angle [tex]\(\theta = \frac{11\pi}{6}\)[/tex].
1. Measure of [tex]\(\theta\)[/tex] in degrees:
[tex]\(\theta\)[/tex] is given in radians as [tex]\(\frac{11\pi}{6}\)[/tex]. Converting it to degrees:
[tex]\[
\theta \text{ (degrees)} = \left( \frac{11\pi}{6} \right) \times \frac{180}{\pi} = 330^\circ
\][/tex]
So, the angle [tex]\(\theta\)[/tex] in degrees is [tex]\(330^\circ\)[/tex].
2. Reference Angle:
The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis. For [tex]\(\theta = 330^\circ\)[/tex]:
[tex]\[
\text{Reference Angle} = 360^\circ - 330^\circ = 30^\circ
\][/tex]
So, the reference angle is [tex]\(30^\circ\)[/tex].
3. Sine and Cosine of [tex]\(\theta\)[/tex]:
Using the previous results to verify sine and cosine:
[tex]\[
\sin \left( \frac{11\pi}{6} \right) = \sin \left( 330^\circ \right) = -\frac{1}{2}
\][/tex]
[tex]\[
\cos \left( \frac{11\pi}{6} \right) = \cos \left( 330^\circ \right) = \frac{\sqrt{3}}{2}
\][/tex]
4. Tangent of [tex]\(\theta\)[/tex]:
Tangent is defined as:
[tex]\[
\tan \left( \frac{11\pi}{6} \right) = \tan \left( 330^\circ \right) = \frac{\sin 330^\circ}{\cos 330^\circ} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\][/tex]
Now, let's go through the given statements:
1. "The measure of the reference angle is [tex]\(30^\circ\)[/tex]."
This is true as computed above.
2. "[tex]\(\sin(\theta) = \frac{i}{2}\)[/tex]"
This is false. The correct value is [tex]\(\sin(330^\circ) = -\frac{1}{2}\)[/tex].
3. "The measure of the reference angle is [tex]\(60^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
4. "[tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] and [tex]\(\tan(\theta) = 1\)[/tex]"
The first part [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] is true but the second part [tex]\(\tan(\theta) = 1\)[/tex] is false. Therefore, the combined statement is false.
5. "The measure of the reference angle is [tex]\(45^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
Hence, the only correct statement from the given options is:
- The measure of the reference angle is [tex]\(30^\circ\)[/tex].
