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To solve for \(\cos 2\alpha\), \(\sin 2\alpha\), and \(\tan 2\alpha\) given that \(\cos \alpha = -\frac{5}{13}\) and \(\pi < \alpha < \frac{3\pi}{2}\), we can use the following trigonometric identities and given information.

1. **Determine \(\sin \alpha\)**:
   \[
   \sin^2 \alpha + \cos^2 \alpha = 1
   \]
   \[
   \sin^2 \alpha + \left(-\frac{5}{13}\right)^2 = 1
   \]
   \[
   \sin^2 \alpha + \frac{25}{169} = 1
   \]
   \[
   \sin^2 \alpha = 1 - \frac{25}{169}
   \]
   \[
   \sin^2 \alpha = \frac{169}{169} - \frac{25}{169}
   \]
   \[
   \sin^2 \alpha = \frac{144}{169}
   \]
   \[
   \sin \alpha = \pm \frac{12}{13}
   \]

   Since \(\alpha\) is in the third quadrant (\(\pi < \alpha < \frac{3\pi}{2}\)), \(\sin \alpha\) is negative:
   \[
   \sin \alpha = -\frac{12}{13}
   \]

2. **Calculate \(\cos 2\alpha\)** using the double-angle identity:
   \[
   \cos 2\alpha = 2\cos^2 \alpha - 1
   \]
   \[
   \cos 2\alpha = 2\left(-\frac{5}{13}\right)^2 - 1
   \]
   \[
   \cos 2\alpha = 2 \cdot \frac{25}{169} - 1
   \]
   \[
   \cos 2\alpha = \frac{50}{169} - 1
   \]
   \[
   \cos 2\alpha = \frac{50}{169} - \frac{169}{169}
   \]
   \[
   \cos 2\alpha = \frac{-119}{169}
   \]

3. **Calculate \(\sin 2\alpha\)** using the double-angle identity:
   \[
   \sin 2\alpha = 2 \sin \alpha \cos \alpha
   \]
   \[
   \sin 2\alpha = 2 \left(-\frac{12}{13}\right) \left(-\frac{5}{13}\right)
   \]
   \[
   \sin 2\alpha = 2 \cdot \frac{60}{169}
   \]
   \[
   \sin 2\alpha = \frac{120}{169}
   \]

4. **Calculate \(\tan 2\alpha\)** using the identity \(\tan 2\alpha = \frac{\sin 2\alpha}{\cos 2\alpha}\):
   \[
   \tan 2\alpha = \frac{\sin 2\alpha}{\cos 2\alpha} = \frac{\frac{120}{169}}{\frac{-119}{169}}
   \]
   \[
   \tan 2\alpha = \frac{120}{-119}
   \]
   \[
   \tan 2\alpha = -\frac{120}{119}
   \]

Therefore, the values are:
\[
\cos 2\alpha = \frac{-119}{169}, \quad \sin 2\alpha = \frac{120}{169}, \quad \tan 2\alpha = -\frac{120}{119}
\]

Would you like details or have any questions about these calculations? Here are five additional questions related to this topic:

1. How do you determine the signs of trigonometric functions in different quadrants?
2. What are the other forms of double-angle identities for \(\cos 2\alpha\)?
3. How can you verify the value of \(\tan 2\alpha\) using its definition in terms of sine and cosine?
4. Can you derive the values of \(\cos 3\alpha\), \(\sin 3\alpha\), and \(\tan 3\alpha\) given \(\cos \alpha\)?
5. How do the sum and difference formulas for sine and cosine apply to this problem?

**Tip:** Remember that knowing the signs of trigonometric functions in each quadrant is crucial for solving trigonometric identities accurately.

cos 2α, sin 2α, tan 2α when cos α = -5/13 and π < α < 3π/2.

Solve for cos 2α, sin 2α, and tan 2α when cos α = -5/13 and π < α < 3π/2. This solution involves the use of double angle identities and Pythagorean theorem. Ideal for students in Grades 11-12.

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