Answer :
To determine which of the given statements about [tex]\(\theta = \frac{11\pi}{6}\)[/tex] are true, let's analyze each statement step-by-step.
1. The measure of the reference angle is [tex]\(30^{\circ}\)[/tex]:
- The angle [tex]\(\theta = \frac{11\pi}{6}\)[/tex] lies in the fourth quadrant.
- The reference angle, which is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis, is calculated as [tex]\(360^{\circ} - 330^{\circ} = 30^{\circ}\)[/tex].
- Therefore, the reference angle is [tex]\(30^{\circ}\)[/tex].
- This statement is true.
2. The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]:
- We already determined that the reference angle for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] is [tex]\(30^{\circ}\)[/tex].
- Thus, the reference angle is not [tex]\(45^{\circ}\)[/tex].
- This statement is false.
3. [tex]\(\tan(\theta) = 1\)[/tex]:
- [tex]\(\tan(\theta)\)[/tex] for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] would require specific calculation or assessment.
- For [tex]\(\tan(\theta)\)[/tex] to be [tex]\(1\)[/tex], [tex]\(\theta\)[/tex] would have to be an angle where the sine and cosine values are equal in magnitude, such as [tex]\(\theta = \frac{\pi}{4}\)[/tex] or [tex]\(\theta = \frac{5\pi}{4}\)[/tex], which is not the case here.
- This statement is false.
4. [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]:
- Evaluating [tex]\(\cos(\frac{11\pi}{6})\)[/tex], we find that it indeed matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex] but with the necessary adjustment for the quadrant.
- However, in the fourth quadrant, where [tex]\(\frac{11\pi}{6}\)[/tex] is located, the cosine is positive.
- This statement is true.
5. The measure of the reference angle is [tex]\(60^{\circ}\)[/tex]:
- As determined originally, the reference angle is [tex]\(30^{\circ}\)[/tex], not [tex]\(60^{\circ}\)[/tex].
- This statement is false.
6. [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex]:
- Evaluating [tex]\(\sin(\frac{11\pi}{6})\)[/tex], we know it does not align with [tex]\(\frac{1}{2}\)[/tex] exactly but should be evaluated with sign considerations about the quadrant.
- Since [tex]\(\sin(\theta)\)[/tex] for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] would be negative due to being in the fourth quadrant.
- This statement is false.
Based on the analysis, the correct evaluations are:
[tex]\[ \begin{aligned} &\text{The measure of the reference angle is } 30^{\circ}: \textbf{True} \\ &\text{The measure of the reference angle is } 45^{\circ}: \textbf{False} \\ &\tan(\theta) = 1: \textbf{False} \\ &\cos(\theta) = \frac{\sqrt{3}}{2}: \textbf{True} \\ &\text{The measure of the reference angle is } 60^{\circ}: \textbf{False} \\ &\sin(\theta) = \frac{1}{2}: \textbf{False} \end{aligned} \][/tex]
1. The measure of the reference angle is [tex]\(30^{\circ}\)[/tex]:
- The angle [tex]\(\theta = \frac{11\pi}{6}\)[/tex] lies in the fourth quadrant.
- The reference angle, which is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis, is calculated as [tex]\(360^{\circ} - 330^{\circ} = 30^{\circ}\)[/tex].
- Therefore, the reference angle is [tex]\(30^{\circ}\)[/tex].
- This statement is true.
2. The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]:
- We already determined that the reference angle for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] is [tex]\(30^{\circ}\)[/tex].
- Thus, the reference angle is not [tex]\(45^{\circ}\)[/tex].
- This statement is false.
3. [tex]\(\tan(\theta) = 1\)[/tex]:
- [tex]\(\tan(\theta)\)[/tex] for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] would require specific calculation or assessment.
- For [tex]\(\tan(\theta)\)[/tex] to be [tex]\(1\)[/tex], [tex]\(\theta\)[/tex] would have to be an angle where the sine and cosine values are equal in magnitude, such as [tex]\(\theta = \frac{\pi}{4}\)[/tex] or [tex]\(\theta = \frac{5\pi}{4}\)[/tex], which is not the case here.
- This statement is false.
4. [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]:
- Evaluating [tex]\(\cos(\frac{11\pi}{6})\)[/tex], we find that it indeed matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex] but with the necessary adjustment for the quadrant.
- However, in the fourth quadrant, where [tex]\(\frac{11\pi}{6}\)[/tex] is located, the cosine is positive.
- This statement is true.
5. The measure of the reference angle is [tex]\(60^{\circ}\)[/tex]:
- As determined originally, the reference angle is [tex]\(30^{\circ}\)[/tex], not [tex]\(60^{\circ}\)[/tex].
- This statement is false.
6. [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex]:
- Evaluating [tex]\(\sin(\frac{11\pi}{6})\)[/tex], we know it does not align with [tex]\(\frac{1}{2}\)[/tex] exactly but should be evaluated with sign considerations about the quadrant.
- Since [tex]\(\sin(\theta)\)[/tex] for [tex]\(\theta = \frac{11\pi}{6}\)[/tex] would be negative due to being in the fourth quadrant.
- This statement is false.
Based on the analysis, the correct evaluations are:
[tex]\[ \begin{aligned} &\text{The measure of the reference angle is } 30^{\circ}: \textbf{True} \\ &\text{The measure of the reference angle is } 45^{\circ}: \textbf{False} \\ &\tan(\theta) = 1: \textbf{False} \\ &\cos(\theta) = \frac{\sqrt{3}}{2}: \textbf{True} \\ &\text{The measure of the reference angle is } 60^{\circ}: \textbf{False} \\ &\sin(\theta) = \frac{1}{2}: \textbf{False} \end{aligned} \][/tex]
