Sure, let's solve these problems step-by-step:
### (a) Find an angle between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] that is coterminal with [tex]\(\frac{33\pi}{10}\)[/tex].
1. Understanding Coterminal Angles:
An angle is coterminal with another if they share the same terminal side when drawn in standard position. Coterminal angles can be found by adding or subtracting full rotations ( [tex]\(2\pi \)[/tex] radians) from the given angle.
2. Given Angle:
[tex]\[\frac{33\pi}{10}\][/tex]
3. Convert the Given Angle:
Let's reduce [tex]\(\frac{33\pi}{10}\)[/tex] to an angle within the range [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
[tex]\[
\frac{33\pi}{10} \quad \text{is more than } 2\pi \; (\approx 6.2832)
\][/tex]
4. Subtract a Full Rotation:
[tex]\[
\frac{33\pi}{10} - 2\pi \times k \quad \text{where } k \text{ is an integer}
\][/tex]
Since [tex]\(\frac{33\pi}{10}\)[/tex] is less than [tex]\(4\pi \)[/tex], we need to subtract only one [tex]\(2\pi\)[/tex] rotation:
[tex]\[
\frac{33\pi}{10} - 2\pi = \frac{33\pi}{10} - \frac{20\pi}{10} = \frac{13\pi}{10}
\][/tex]
Here, [tex]\(\frac{13\pi}{10}\)[/tex] is already within the range [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
5. Coterminal Angle:
[tex]\[
\boxed{\frac{13\pi}{10}}
\][/tex]
### (b) Find an angle between [tex]\(0^{\circ}\)[/tex] and [tex]\(360^{\circ}\)[/tex] that is coterminal with [tex]\(930^{\circ}\)[/tex].
1. Understanding Coterminal Angles:
Similarly, for degrees, an angle is coterminal if it is the same as the given angle plus or minus multiples of [tex]\(360^\circ\)[/tex] (a full rotation).
2. Given Angle:
[tex]\(930^\circ\)[/tex]
3. Convert the Given Angle:
Let's reduce [tex]\(930^\circ\)[/tex] to an angle within the range [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
4. Subtract Full Rotations:
[tex]\[
930^\circ \mod 360^\circ
\][/tex]
This means we subtract multiples of [tex]\(360^\circ\)[/tex] until the result is within [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
[tex]\[
930^\circ - 2 \times 360^\circ = 930^\circ - 720^\circ = 210^\circ
\][/tex]
Here [tex]\(210^\circ\)[/tex] is already within the range [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
5. Coterminal Angle:
[tex]\[
\boxed{210^\circ}
\][/tex]
So, the final answers are:
- (a) [tex]\(\frac{13\pi}{10}\)[/tex]
- (b) [tex]\(210^\circ\)[/tex]
