To determine which angle measure is equivalent to the angle [tex]\( \frac{17 \pi}{4} \)[/tex], we follow these steps:
1. Calculate the angle in radians:
The given angle in radians is [tex]\( \frac{17 \pi}{4} \)[/tex].
2. Convert the angle from radians to degrees:
The relationship between radians and degrees is [tex]\( 180^\circ = \pi \)[/tex] radians. Therefore, to convert from radians to degrees, we use the conversion factor [tex]\(\frac{180^\circ}{\pi}\)[/tex].
[tex]\[
\text{Angle in degrees} = \frac{17 \pi}{4} \times \frac{180^\circ}{\pi}
\][/tex]
3. Simplify the expression:
The [tex]\(\pi\)[/tex] in the numerator and denominator cancels out, and we are left with:
[tex]\[
\text{Angle in degrees} = \frac{17 \times 180^\circ}{4}
\][/tex]
Calculate the multiplication and division:
[tex]\[
= \frac{3060^\circ}{4}
= 765^\circ
\][/tex]
4. Find the equivalent angle within the range [0, 360) degrees:
Angles are periodic with a period of 360 degrees. To find an equivalent angle within one full rotation, we take the remainder when 765 degrees is divided by 360 degrees:
[tex]\[
765^\circ \mod 360^\circ
\][/tex]
Perform the division and find the remainder:
[tex]\[
765^\circ = 2 \times 360^\circ + 45^\circ
\][/tex]
Since [tex]\( \frac{765}{360} = 2 \)[/tex] remainder [tex]\( 45 \)[/tex], the equivalent angle is:
[tex]\[
45^\circ
\][/tex]
Therefore, the equivalent angle to [tex]\( \frac{17 \pi}{4} \)[/tex] is [tex]\( 45^\circ \)[/tex]. Thus, the correct answer is:
D. [tex]\( 45^\circ \)[/tex]
