To solve the given problem, we'll address each part step-by-step.
### Part (a)
To find an angle between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] that is coterminal with [tex]\(-\frac{17\pi}{10}\)[/tex], we need to adjust the given angle by adding multiples of [tex]\(2\pi\)[/tex] until the result falls within the desired range.
1. Identify the given angle:
[tex]\[
\text{Given angle} = -\frac{17\pi}{10}
\][/tex]
2. Add [tex]\(2\pi\)[/tex] to the given angle:
[tex]\[
-\frac{17\pi}{10} + 2\pi = -\frac{17\pi}{10} + \frac{20\pi}{10} = \frac{3\pi}{10}
\][/tex]
3. Verify the result:
We see that [tex]\(\frac{3\pi}{10}\)[/tex] is between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex]. So, this is the coterminal angle within the desired range.
Therefore, the angle between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] that is coterminal with [tex]\(-\frac{17\pi}{10}\)[/tex] is:
[tex]\[
0.9424777960769379 \text{ radians}
\][/tex]
### Part (b)
To find an angle between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] that is coterminal with [tex]\(1170^\circ\)[/tex], we need to reduce the given angle by subtracting [tex]\(360^\circ\)[/tex] until it falls within the desired range.
1. Identify the given angle:
[tex]\[
\text{Given angle} = 1170^\circ
\][/tex]
2. Find the remainder when the given angle is divided by [tex]\(360^\circ\)[/tex]:
[tex]\[
1170^\circ \mod 360^\circ = 1170 - \left\lfloor \frac{1170}{360} \right\rfloor \times 360
\][/tex]
3. Perform the division:
[tex]\[
\left\lfloor \frac{1170}{360} \right\rfloor = 3 \quad \text{(the greatest integer less than or equal to the quotient)}
\][/tex]
[tex]\[
1170 - 3 \times 360 = 1170 - 1080 = 90
\][/tex]
4. Verify the result:
We see that [tex]\(90^\circ\)[/tex] is between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex]. So, this is the coterminal angle within the desired range.
Therefore, the angle between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] that is coterminal with [tex]\(1170^\circ\)[/tex] is:
[tex]\[
90^\circ
\][/tex]
