To find the value of [tex]\(\operatorname{cosec}\left(-\frac{19 \pi}{3}\right)\)[/tex], we can follow these steps:
1. Simplify the Angle:
The angle [tex]\(-\frac{19 \pi}{3}\)[/tex] can be simplified by recognizing that trigonometric functions are periodic with a period of [tex]\(2\pi\)[/tex]. Therefore, we can add or subtract multiples of [tex]\(2\pi\)[/tex] to bring the angle within a standard range [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
[tex]\[
-\frac{19 \pi}{3} + 2\pi \cdot 6 = -\frac{19 \pi}{3} + \frac{12 \pi}{1} = -\frac{19 \pi}{3} + \frac{24 \pi}{3} = \frac{5 \pi}{3}
\][/tex]
2. Calculate the Sine of the Angle:
Now we need to compute [tex]\(\sin\left(\frac{5 \pi}{3}\right)\)[/tex].
The angle [tex]\(\frac{5 \pi}{3}\)[/tex] lies in the fourth quadrant of the unit circle where sine is negative. Since [tex]\(\frac{5 \pi}{3}\)[/tex] is equivalent to [tex]\(2\pi - \frac{\pi}{3}\)[/tex]:
[tex]\[
\sin\left(2\pi - \frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)
\][/tex]
We know that [tex]\(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)[/tex].
Therefore:
[tex]\[
\sin\left(\frac{5 \pi}{3}\right) = -\frac{\sqrt{3}}{2}
\][/tex]
3. Calculate the Cosecant Value:
The cosecant function is the reciprocal of the sine function. Hence, we calculate:
[tex]\[
\operatorname{cosec}\left(\frac{5 \pi}{3}\right) = \frac{1}{\sin\left(\frac{5 \pi}{3}\right)}
\][/tex]
Substituting the value found for the sine:
[tex]\[
\operatorname{cosec}\left(\frac{5 \pi}{3}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}
\][/tex]
To rationalize the denominator:
[tex]\[
-\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
\][/tex]
4. Final Answer:
Therefore, the value of [tex]\(\operatorname{cosec}\left(-\frac{19 \pi}{3}\right)\)[/tex] is:
[tex]\[
-\frac{2\sqrt{3}}{3} \approx -1.1547005383792528
\][/tex]
