Answer :
Let's analyze the problem where the angle [tex]\(\theta\)[/tex] is [tex]\(\frac{5 \pi}{4}\)[/tex]. We need to determine which of the given statements are true.
1. The measure of the reference angle is [tex]\(45^{\circ}\)[/tex].
Since [tex]\(\theta = \frac{5 \pi}{4}\)[/tex] is located in the third quadrant, the reference angle is calculated by subtracting [tex]\(\pi\)[/tex] from [tex]\(\theta\)[/tex]:
[tex]\[ \text{Reference angle} = \theta - \pi = \frac{5 \pi}{4} - \pi = \frac{5 \pi}{4} - \frac{4 \pi}{4} = \frac{\pi}{4} \][/tex]
Converting this reference angle to degrees:
[tex]\[ \frac{\pi}{4} \times \frac{180^{\circ}}{\pi} = 45^{\circ} \][/tex]
Thus, the statement "The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]" is true.
2. [tex]\(\tan(\theta) = 1\)[/tex]
The tangent function for an angle in the third quadrant is positive and for [tex]\(\theta = \frac{5 \pi}{4}\)[/tex], the value is:
[tex]\[ \tan\left(\frac{5 \pi}{4}\right) = 1 \][/tex]
Therefore, the statement "[tex]\(\tan(\theta) = 1\)[/tex]" is true.
3. The measure of the reference angle is [tex]\(30^{\circ}\)[/tex].
From our previous calculation, we found that the reference angle is [tex]\(45^{\circ}\)[/tex]. Therefore, the statement "The measure of the reference angle is [tex]\(30^{\circ}\)[/tex]" is false.
4. [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex]
For [tex]\(\theta = \frac{5 \pi}{4}\)[/tex] in the third quadrant:
[tex]\[ \sin\left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
Therefore, the statement "[tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is false.
5. [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex]
Similarly, for the cosine function in the third quadrant:
[tex]\[ \cos\left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
Therefore, the statement "[tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is false.
6. The measure of the reference angle is [tex]\(60^{\circ}\)[/tex].
Again, from our previous reference angle calculation, we know the reference angle is [tex]\(45^{\circ}\)[/tex]. Therefore, the statement "The measure of the reference angle is [tex]\(60^{\circ}\)[/tex]" is false.
Summarizing the true statements:
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex].
- [tex]\(\tan(\theta) = 1\)[/tex].
1. The measure of the reference angle is [tex]\(45^{\circ}\)[/tex].
Since [tex]\(\theta = \frac{5 \pi}{4}\)[/tex] is located in the third quadrant, the reference angle is calculated by subtracting [tex]\(\pi\)[/tex] from [tex]\(\theta\)[/tex]:
[tex]\[ \text{Reference angle} = \theta - \pi = \frac{5 \pi}{4} - \pi = \frac{5 \pi}{4} - \frac{4 \pi}{4} = \frac{\pi}{4} \][/tex]
Converting this reference angle to degrees:
[tex]\[ \frac{\pi}{4} \times \frac{180^{\circ}}{\pi} = 45^{\circ} \][/tex]
Thus, the statement "The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]" is true.
2. [tex]\(\tan(\theta) = 1\)[/tex]
The tangent function for an angle in the third quadrant is positive and for [tex]\(\theta = \frac{5 \pi}{4}\)[/tex], the value is:
[tex]\[ \tan\left(\frac{5 \pi}{4}\right) = 1 \][/tex]
Therefore, the statement "[tex]\(\tan(\theta) = 1\)[/tex]" is true.
3. The measure of the reference angle is [tex]\(30^{\circ}\)[/tex].
From our previous calculation, we found that the reference angle is [tex]\(45^{\circ}\)[/tex]. Therefore, the statement "The measure of the reference angle is [tex]\(30^{\circ}\)[/tex]" is false.
4. [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex]
For [tex]\(\theta = \frac{5 \pi}{4}\)[/tex] in the third quadrant:
[tex]\[ \sin\left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
Therefore, the statement "[tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is false.
5. [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex]
Similarly, for the cosine function in the third quadrant:
[tex]\[ \cos\left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
Therefore, the statement "[tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is false.
6. The measure of the reference angle is [tex]\(60^{\circ}\)[/tex].
Again, from our previous reference angle calculation, we know the reference angle is [tex]\(45^{\circ}\)[/tex]. Therefore, the statement "The measure of the reference angle is [tex]\(60^{\circ}\)[/tex]" is false.
Summarizing the true statements:
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex].
- [tex]\(\tan(\theta) = 1\)[/tex].
