To solve for the degree measure, reference angle in degrees, and reference angle in radians for [tex]\(\theta = \frac{7\pi}{4}\)[/tex], let's go through the steps in detail.
### Step 1: Convert the angle from radians to degrees.
The conversion formula from radians to degrees is:
[tex]\[
\text{Degrees} = \theta \times \frac{180}{\pi}
\][/tex]
Given [tex]\(\theta = \frac{7\pi}{4}\)[/tex]:
[tex]\[
\theta_{\text{deg}} = \frac{7\pi}{4} \times \frac{180}{\pi} = 7 \times \frac{180}{4} = 7 \times 45 = 315 \text{ degrees}
\][/tex]
So, the degree measure is:
[tex]\[
\theta_{\text{deg}} = 315 \text{ degrees}
\][/tex]
### Step 2: Determine the reference angle in degrees.
The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. For angles in different quadrants, the reference angle is found differently. Since 315 degrees is in the fourth quadrant (as it is between 270 degrees and 360 degrees), the reference angle is calculated as:
[tex]\[
\text{Reference angle} = 360^\circ - \theta_{\text{deg}}
\][/tex]
Substituting the value of [tex]\(\theta_{\text{deg}} = 315^\circ\)[/tex]:
[tex]\[
\text{Reference angle} = 360^\circ - 315^\circ = 45^\circ
\][/tex]
So, the reference angle in degrees is:
[tex]\[
45 \text{ degrees}
\][/tex]
### Step 3: Convert the reference angle from degrees to radians.
The conversion formula from degrees to radians is:
[tex]\[
\text{Radians} = \text{Degrees} \times \frac{\pi}{180}
\][/tex]
Given the reference angle is 45 degrees:
[tex]\[
\text{Reference angle in radians} = 45 \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4}
\][/tex]
Thus, the reference angle in radians is:
[tex]\[
0.7853981633974483 \text{ radians} \left( \approx \frac{\pi}{4} \right)
\][/tex]
### Summary
1. Degree measure: [tex]\(315\)[/tex] degrees
2. Reference angle in degrees: [tex]\(45\)[/tex] degrees
3. Reference angle in radians: [tex]\(0.7853981633974483\)[/tex] radians
