Sure! Let's solve this step-by-step.
Given:
[tex]\[ \csc \theta = -\frac{6}{5} \][/tex]
We are also given that \( \pi < \theta < \frac{3\pi}{2} \).
1. Find \(\sin \theta\):
The cosecant function, \(\csc \theta\), is the reciprocal of the sine function:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Therefore,
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{-\frac{6}{5}} = -\frac{5}{6}
\][/tex]
2. Find \(\cos \theta\):
To find \(\cos \theta\), we can use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
We already have \(\sin \theta = -\frac{5}{6}\), so:
\begin{align}
\left( -\frac{5}{6} \right)^2 + \cos^2 \theta &= 1 \\
\frac{25}{36} + \cos^2 \theta &= 1 \\
\cos^2 \theta &= 1 - \frac{25}{36} \\
\cos^2 \theta &= \frac{36}{36} - \frac{25}{36} \\
\cos^2 \theta &= \frac{11}{36}
\end{align}
Since \( \pi < \theta < \frac{3\pi}{2} \) (which lies in the third quadrant where \(\cos \theta\) is negative):
[tex]\[
\cos \theta = -\sqrt{\frac{11}{36}} = -\frac{\sqrt{11}}{6} \approx -0.553
\][/tex]
3. Find \(\sin (2 \theta)\):
Using the double angle formula for sine:
[tex]\[
\sin (2 \theta) = 2 \sin \theta \cos \theta
\][/tex]
Substituting the values we found:
[tex]\[
\sin (2 \theta) = 2 \left(-\frac{5}{6}\right) \left(-\frac{\sqrt{11}}{6}\right) = 2 \cdot \frac{5\sqrt{11}}{36} = \frac{10\sqrt{11}}{36} = \frac{5\sqrt{11}}{18} \approx 0.921
\][/tex]
4. Find \(\cos (2 \theta)\):
Using the double angle formula for cosine:
[tex]\[
\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta
\][/tex]
Substituting the values we have:
\begin{align}
\cos^2 \theta &= \frac{11}{36}, \\
\sin^2 \theta &= \left(-\frac{5}{6}\right)^2 = \frac{25}{36}, \\
\cos (2 \theta) &= \frac{11}{36} - \frac{25}{36} = -\frac{14}{36} = -\frac{7}{18} \approx -0.389
\end{align}
5. Find \(\tan (2 \theta)\):
The tangent of the double angle can be found using the identity:
[tex]\[
\tan (2 \theta) = \frac{\sin (2 \theta)}{\cos (2 \theta)}
\][/tex]
Substituting the values:
[tex]\[
\tan (2 \theta) = \frac{\frac{5\sqrt{11}}{18}}{-\frac{7}{18}} = \frac{5\sqrt{11}}{-7} \approx -2.369
\][/tex]
To summarize:
- \(\sin (2 \theta) \approx 0.921\)
- \(\cos (2 \theta) \approx -0.389\)
- [tex]\(\tan (2 \theta) \approx -2.369\)[/tex]
