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Sagot :
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]
### 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]
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