Answer :
Let's analyze the given information and determine which statements are true for the angle [tex]\(\theta = \frac{11\pi}{6}\)[/tex].
### Step 1: Determine the Reference Angle
Given [tex]\(\theta = \frac{11\pi}{6}\)[/tex], we need to find the reference angle. The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis.
To find the reference angle for [tex]\(\theta = \frac{11\pi}{6}\)[/tex], we need to compute the absolute difference between [tex]\(\theta\)[/tex] and [tex]\(2\pi\)[/tex]:
[tex]\[ \text{Reference Angle (in radians)} = | \theta - 2\pi | = \left| \frac{11\pi}{6} - 2\pi \right| \][/tex]
Compute [tex]\(2\pi\)[/tex] in terms of [tex]\(\frac{\pi}{6}\)[/tex]:
[tex]\[ 2\pi = \frac{12\pi}{6} \][/tex]
Then:
[tex]\[ \left| \frac{11\pi}{6} - \frac{12\pi}{6} \right| = \left| \frac{11\pi}{6} - \frac{12\pi}{6} \right| = \left| \frac{-\pi}{6} \right| = \frac{\pi}{6} \][/tex]
Thus, the reference angle in radians is [tex]\(\frac{\pi}{6}\)[/tex].
Now, convert this to degrees:
[tex]\[ \text{Reference Angle (in degrees)} = \frac{\pi}{6} \cdot \frac{180^\circ}{\pi} = 30^\circ \][/tex]
So, the reference angle is [tex]\(30^\circ\)[/tex].
### Step 2: Evaluate Trigonometric Functions
#### Evaluate [tex]\(\sin(\theta)\)[/tex]:
[tex]\(\theta = \frac{11\pi}{6}\)[/tex] lies in the fourth quadrant, where sine values are negative. The sine of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex], so:
[tex]\[ \sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2} \][/tex]
#### Evaluate [tex]\(\cos(\theta)\)[/tex]:
In the fourth quadrant, cosine values are positive. The cosine of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is:
[tex]\[ \cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]
#### Evaluate [tex]\(\tan(\theta)\)[/tex]:
In the fourth quadrant, tangent values are negative. The tangent of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is:
[tex]\[ \tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}} \][/tex]
### Step 3: Verify Statements
Let's verify each given statement:
1. The measure of the reference angle is [tex]\(60^\circ\)[/tex].
- False. The reference angle is [tex]\(30^\circ\)[/tex], not [tex]\(60^\circ\)[/tex].
2. [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex]
- False. [tex]\(\sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2}\)[/tex].
3. [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]
- True. [tex]\(\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}\)[/tex].
4. The measure of the reference angle is [tex]\(30^\circ\)[/tex].
- True. The reference angle is indeed [tex]\(30^\circ\)[/tex].
5. [tex]\(\tan(\theta) = 1\)[/tex]
- False. [tex]\(\tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}}\)[/tex].
6. The measure of the reference angle is [tex]\(45^\circ\)[/tex].
- False. The reference angle is [tex]\(30^\circ\)[/tex].
### Conclusion
The correct statements are:
- [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(30^\circ\)[/tex]
### Step 1: Determine the Reference Angle
Given [tex]\(\theta = \frac{11\pi}{6}\)[/tex], we need to find the reference angle. The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis.
To find the reference angle for [tex]\(\theta = \frac{11\pi}{6}\)[/tex], we need to compute the absolute difference between [tex]\(\theta\)[/tex] and [tex]\(2\pi\)[/tex]:
[tex]\[ \text{Reference Angle (in radians)} = | \theta - 2\pi | = \left| \frac{11\pi}{6} - 2\pi \right| \][/tex]
Compute [tex]\(2\pi\)[/tex] in terms of [tex]\(\frac{\pi}{6}\)[/tex]:
[tex]\[ 2\pi = \frac{12\pi}{6} \][/tex]
Then:
[tex]\[ \left| \frac{11\pi}{6} - \frac{12\pi}{6} \right| = \left| \frac{11\pi}{6} - \frac{12\pi}{6} \right| = \left| \frac{-\pi}{6} \right| = \frac{\pi}{6} \][/tex]
Thus, the reference angle in radians is [tex]\(\frac{\pi}{6}\)[/tex].
Now, convert this to degrees:
[tex]\[ \text{Reference Angle (in degrees)} = \frac{\pi}{6} \cdot \frac{180^\circ}{\pi} = 30^\circ \][/tex]
So, the reference angle is [tex]\(30^\circ\)[/tex].
### Step 2: Evaluate Trigonometric Functions
#### Evaluate [tex]\(\sin(\theta)\)[/tex]:
[tex]\(\theta = \frac{11\pi}{6}\)[/tex] lies in the fourth quadrant, where sine values are negative. The sine of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex], so:
[tex]\[ \sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2} \][/tex]
#### Evaluate [tex]\(\cos(\theta)\)[/tex]:
In the fourth quadrant, cosine values are positive. The cosine of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is:
[tex]\[ \cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]
#### Evaluate [tex]\(\tan(\theta)\)[/tex]:
In the fourth quadrant, tangent values are negative. The tangent of the reference angle [tex]\(\frac{\pi}{6}\)[/tex] is:
[tex]\[ \tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}} \][/tex]
### Step 3: Verify Statements
Let's verify each given statement:
1. The measure of the reference angle is [tex]\(60^\circ\)[/tex].
- False. The reference angle is [tex]\(30^\circ\)[/tex], not [tex]\(60^\circ\)[/tex].
2. [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex]
- False. [tex]\(\sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2}\)[/tex].
3. [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]
- True. [tex]\(\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}\)[/tex].
4. The measure of the reference angle is [tex]\(30^\circ\)[/tex].
- True. The reference angle is indeed [tex]\(30^\circ\)[/tex].
5. [tex]\(\tan(\theta) = 1\)[/tex]
- False. [tex]\(\tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}}\)[/tex].
6. The measure of the reference angle is [tex]\(45^\circ\)[/tex].
- False. The reference angle is [tex]\(30^\circ\)[/tex].
### Conclusion
The correct statements are:
- [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(30^\circ\)[/tex]
