Let's solve this step-by-step:
1. Measure of Angle [tex]\( \theta \)[/tex] in Degrees:
- Given [tex]\( \theta = \frac{11\pi}{6} \)[/tex] radians.
- To convert [tex]\( \theta \)[/tex] to degrees, we multiply by [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\theta_{\text{deg}} = \frac{11\pi}{6} \times \frac{180}{\pi} = \frac{11 \times 180}{6} = 330^{\circ}
\][/tex]
2. Reference Angle:
- The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis.
- Since [tex]\( \theta = 330^{\circ} \)[/tex] is in the fourth quadrant, the reference angle is:
[tex]\[
360^{\circ} - 330^{\circ} = 30^{\circ}
\][/tex]
3. Calculating Trigonometric Values:
- Sine:
[tex]\[
\sin(330^{\circ}) = \sin(360^{\circ} - 30^{\circ}) = -\sin(30^{\circ}) = -\frac{1}{2}
\][/tex]
- Cosine:
[tex]\[
\cos(330^{\circ}) = \cos(360^{\circ} - 30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}
\][/tex]
- Tangent:
[tex]\[
\tan(330^{\circ}) = \tan(360^{\circ} - 30^{\circ}) = -\tan(30^{\circ}) = -\frac{1}{\sqrt{3}}
\][/tex]
4. Check the Statements:
- The measure of the reference angle is [tex]\( 60^{\circ} \)[/tex]: This is false; the reference angle is [tex]\( 30^{\circ} \)[/tex].
- [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex]: This is false; [tex]\(\sin(330^{\circ}) = -\frac{1}{2}\)[/tex].
- [tex]\(\tan(\theta) = 1\)[/tex]: This is false; [tex]\(\tan(330^{\circ}) = -\frac{1}{\sqrt{3}}\)[/tex].
- The measure of the reference angle is [tex]\( 45^{\circ} \)[/tex]: This is false; the reference angle is [tex]\( 30^{\circ} \)[/tex].
- [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]: This is true; [tex]\(\cos(330^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex].
- The measure of the reference angle is [tex]\( 30^{\circ} \)[/tex]: This is true.
Therefore, the correct statements are:
[tex]\[
\boxed{\cos (\theta)=\frac{\sqrt{3}}{2}} \quad \text{and} \quad \boxed{\text{The measure of the reference angle is } 30^{\circ}.}
\][/tex]
