To find a standard angle coterminal with [tex]\(\theta = \frac{17\pi}{6}\)[/tex] and evaluate [tex]\(\csc \theta\)[/tex], follow these steps:
### Step 1: Find a Standard Angle Coterminal with [tex]\(\theta = \frac{17\pi}{6}\)[/tex]
1. Convert the initial angle into a standard position:
An angle in standard position is typically within the range of [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex] radians. To find this angle, reduce it by subtracting multiples of [tex]\(2\pi\)[/tex].
We know that [tex]\(2\pi = \frac{12\pi}{6}\)[/tex]. We can use this to reduce [tex]\(\frac{17\pi}{6}\)[/tex]:
[tex]\[
\frac{17\pi}{6} \equiv \frac{17\pi}{6} \mod 2\pi = \frac{17\pi}{6} \mod \frac{12\pi}{6} = \frac{17\pi}{6} - \left(\left\lfloor \frac{17}{12} \right\rfloor \times \frac{12\pi}{6}\right)
\][/tex]
Calculate the integer division of [tex]\(17\)[/tex] by [tex]\(12\)[/tex]:
[tex]\[
\left\lfloor \frac{17}{12} \right\rfloor = 1
\][/tex]
Subtract [tex]\(2\pi\)[/tex] (which is [tex]\(\frac{12\pi}{6}\)[/tex]) from [tex]\(\frac{17\pi}{6}\)[/tex]:
[tex]\[
\frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}
\][/tex]
Thus, a standard angle coterminal with [tex]\(\frac{17\pi}{6}\)[/tex] is [tex]\(\frac{5\pi}{6}\)[/tex]. However, what we want is the coterminal angle within one full rotation of [tex]\(2\pi\)[/tex], not just a reduction. Based on the standard angle property, further calculations lead us to:
[tex]\[
\frac{17\pi}{6} \mod 2\pi = 2.617\ldots\, (closely equivalent to \frac{5\pi}{6} + 2)
\][/tex]
### Step 2: Evaluate [tex]\(\csc \theta\)[/tex] for the Standard Angle
1. Determine the sine of the standard angle:
Given the angle [tex]\(\theta = 2.617993877991495\)[/tex] radians, which corresponds to approximately [tex]\(\frac{5\pi}{6}\)[/tex], you can use the unit circle to find [tex]\(\sin \left(\frac{5\pi}{6}\right)\)[/tex].
For [tex]\(\frac{5\pi}{6}\)[/tex], its sine is the same as:
[tex]\[
\sin \left(\frac{5\pi}{6}\right) = \sin \left(\pi - \frac{\pi}{6}\right) = \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}
\][/tex]
The reference angle (unit circle property):
[tex]\[
\theta = 2.617 \text{ approximately}
\][/tex]
2. Calculate the cosecant:
[tex]\[
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{0.5} = 2.0000000000000018
\][/tex]
### Conclusion
The standard angle coterminal with [tex]\(\theta = \frac{17\pi}{6}\)[/tex] is approximately [tex]\(2.617993877991495\)[/tex] radians. Evaluating [tex]\(\csc \theta\)[/tex] for this standard angle yields approximately [tex]\(2.0000000000000018\)[/tex].
