Let's solve the problem step-by-step.
Given:
[tex]\[
\sin \left( \frac{19\pi}{2} \right)
\][/tex]
First, we need to reduce the given angle to find its equivalent angle within one full circle. A full circle in radians is [tex]\(2\pi\)[/tex]. Therefore, we start by finding the equivalent angle within one full circle by removing full circles (multiples of [tex]\(2\pi\)[/tex]).
[tex]\[
\frac{19\pi}{2} \mod 2\pi
\][/tex]
Since [tex]\(2\pi\)[/tex] is equivalent to [tex]\(4\pi/2\)[/tex], let's express [tex]\(2\pi\)[/tex] as [tex]\(4\pi/2\)[/tex]:
[tex]\[
\frac{19\pi}{2} \mod \frac{4\pi}{2}
\][/tex]
To find this, we compute:
[tex]\[
\frac{19\pi}{2} \div \frac{4\pi}{2} = \frac{19\pi}{2} \times \frac{2}{4\pi} = \frac{19}{4} = 4 \text{ remainder } \frac{3}{4}
\][/tex]
This means:
[tex]\[
\frac{19\pi}{2} = 4 \cdot 2\pi + \frac{3\pi}{2}
\][/tex]
So, [tex]\(\frac{19\pi}{2}\)[/tex] is equivalent to:
[tex]\[
\frac{3\pi}{2}
\][/tex]
Now, the reference angle is the smallest positive angle formed with the x-axis, which can be found by examining the equivalent angle within one circle (0 to [tex]\(2\pi\)[/tex]):
Since [tex]\(\frac{3\pi}{2}\)[/tex] is in the third quadrant, the reference angle is:
[tex]\[
\pi - \frac{3\pi}{2} = \frac{\pi}{2}
\][/tex]
Therefore, the reference angle for [tex]\(\frac{19\pi}{2}\)[/tex] is:
[tex]\[
\frac{\pi}{2}
\][/tex]
Now, evaluating the sine of [tex]\(\frac{19\pi}{2}\)[/tex]:
Considering [tex]\( \sin \left( \frac{19\pi}{2} \right) \)[/tex] is equivalent to [tex]\( \sin \left( \frac{3\pi}{2} \right) \)[/tex], and using the property of the sine function:
[tex]\[
\sin \left( \frac{3\pi}{2} \right) = -1
\][/tex]
Thus,
[tex]\[
\sin \left( \frac{19\pi}{2} \right) = -1
\][/tex]
So, the exact answer is:
[tex]\[
\sin \left( \frac{19\pi}{2} \right) = -1
\][/tex]
