Answer :
To solve the problem, we need to evaluate the trigonometric functions for the angle [tex]\(\theta = \frac{\pi}{4}\)[/tex] and determine the reference angle.
### Step-by-Step Solution:
1. Calculate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]
Thus, the statement "[tex]\(\tan (\theta) = 1\)[/tex]" is true.
2. Determine the Reference Angle:
The reference angle for [tex]\(\theta = \frac{\pi}{4}\)[/tex] can be converted to degrees:
[tex]\[ \frac{\pi}{4} \text{ radians} = 45^{\circ} \][/tex]
So, the statements "[tex]\(\text{The measure of the reference angle is } 45^{\circ}\)[/tex]" is true.
The other potential measures of reference angles 30°, and 60° are not correct in this case.
3. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.
4. Calculate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.
### Summary:
Based on the calculations and evaluations, the true statements are:
- [tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]
Thus the correct answers are:
- [tex]\(\cos (\theta)=\frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta)=\frac{\sqrt{2}}{2}\)[/tex]
Therefore, the answers are [tex]\( [5, 3] \)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]
Thus, the statement "[tex]\(\tan (\theta) = 1\)[/tex]" is true.
2. Determine the Reference Angle:
The reference angle for [tex]\(\theta = \frac{\pi}{4}\)[/tex] can be converted to degrees:
[tex]\[ \frac{\pi}{4} \text{ radians} = 45^{\circ} \][/tex]
So, the statements "[tex]\(\text{The measure of the reference angle is } 45^{\circ}\)[/tex]" is true.
The other potential measures of reference angles 30°, and 60° are not correct in this case.
3. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.
4. Calculate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.
### Summary:
Based on the calculations and evaluations, the true statements are:
- [tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]
Thus the correct answers are:
- [tex]\(\cos (\theta)=\frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta)=\frac{\sqrt{2}}{2}\)[/tex]
Therefore, the answers are [tex]\( [5, 3] \)[/tex].
