To solve for \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) given that \(\sec \theta = -3\) and \(90^\circ < \theta < 180^\circ\), we can follow these steps:
1. Determine \(\cos \theta\):
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Given \(\sec \theta = -3\), we have:
[tex]\[
\cos \theta = \frac{1}{\sec \theta} = \frac{1}{-3} = -\frac{1}{3}
\][/tex]
2. Determine \(\sin \theta\):
Using the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substituting \(\cos \theta = -\frac{1}{3}\), we get:
[tex]\[
\sin^2 \theta + \left(-\frac{1}{3}\right)^2 = 1
\][/tex]
[tex]\[
\sin^2 \theta + \frac{1}{9} = 1
\][/tex]
[tex]\[
\sin^2 \theta = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}
\][/tex]
Taking the positive square root (since \(\sin \theta\) is positive in the second quadrant):
[tex]\[
\sin \theta = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}
\][/tex]
3. Calculate \(\sin 2\theta\):
Using the double-angle formula for sine:
[tex]\[
\sin 2\theta = 2 \sin \theta \cos \theta
\][/tex]
Substituting the values for \(\sin \theta\) and \(\cos \theta\):
[tex]\[
\sin 2\theta = 2 \left(\frac{2\sqrt{2}}{3}\right) \left(-\frac{1}{3}\right) = 2 \left(-\frac{2\sqrt{2}}{9}\right) = -\frac{4\sqrt{2}}{9}
\][/tex]
4. Calculate \(\cos 2\theta\):
Using the double-angle formula for cosine:
[tex]\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\][/tex]
Substituting the already known values for \(\cos \theta\) and \(\sin \theta\):
[tex]\[
\cos 2\theta = \left(-\frac{1}{3}\right)^2 - \left(\frac{2\sqrt{2}}{3}\right)^2 = \frac{1}{9} - \frac{8}{9} = -\frac{7}{9}
\][/tex]
5. Calculate \(\tan 2\theta\):
Using the relationship:
[tex]\[
\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}
\][/tex]
Substituting the calculated values for \(\sin 2\theta\) and \(\cos 2\theta\):
[tex]\[
\tan 2\theta = \frac{-\frac{4\sqrt{2}}{9}}{-\frac{7}{9}} = \frac{4\sqrt{2}}{7}
\][/tex]
Thus, the exact values are:
[tex]\[
\sin 2\theta = -\frac{4\sqrt{2}}{9},
\][/tex]
[tex]\[
\cos 2\theta = -\frac{7}{9},
\][/tex]
[tex]\[
\tan 2\theta = \frac{4\sqrt{2}}{7}
\][/tex]
