Given that $\sin \theta = -\frac{4}{9}$ and $\theta$ is in the third quadrant, we need to find the exact values of $\sec \theta$ and $\tan \theta$.

Step 1: Determine $\cos \theta$

We know that: $\sin^2 \theta + \cos^2 \theta = 1$ Substitute $\sin \theta = -\frac{4}{9}$: $\left(-\frac{4}{9}\right)^2 + \cos^2 \theta = 1$ $\frac{16}{81} + \cos^2 \theta = 1$ $\cos^2 \theta = 1 - \frac{16}{81}$ $\cos^2 \theta = \frac{81}{81} - \frac{16}{81} = \frac{65}{81}$ Since $\theta$ is in the third quadrant, $\cos \theta$ is negative: $\cos \theta = -\sqrt{\frac{65}{81}} = -\frac{\sqrt{65}}{9}$

Step 2: Calculate $\sec \theta$

The secant is the reciprocal of the cosine: $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{65}}{9}} = -\frac{9}{\sqrt{65}}$ To rationalize the denominator: $\sec \theta = -\frac{9}{\sqrt{65}} \times \frac{\sqrt{65}}{\sqrt{65}} = -\frac{9\sqrt{65}}{65}$

Step 3: Calculate $\tan \theta$

The tangent is the ratio of sine to cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{9}}{-\frac{\sqrt{65}}{9}} = \frac{-4}{-\sqrt{65}} = \frac{4}{\sqrt{65}}$ Again, rationalizing the denominator: $\tan \theta = \frac{4}{\sqrt{65}} \times \frac{\sqrt{65}}{\sqrt{65}} = \frac{4\sqrt{65}}{65}$

Final Answers:

$\sec \theta = -\frac{9\sqrt{65}}{65}, \quad \tan \theta = \frac{4\sqrt{65}}{65}$

Would you like more details on any of these steps, or do you have further questions?

Here are some related questions to consider:

1. What is the significance of the quadrant in determining the signs of trigonometric functions?
2. How do you rationalize a denominator, and why is it done?
3. How can you derive the Pythagorean identity from the unit circle?
4. What are the other trigonometric identities that can be useful in problems like this?
5. How do you determine the reference angle for a given angle in any quadrant?
6. What are the steps to convert an angle in radians to degrees and vice versa?
7. How does the CAST rule help in determining the sign of trigonometric functions?
8. How can you graphically represent the values of $\sin$, $\cos$, and $\tan$ for angles in different quadrants?

Tip: Remember that understanding the unit circle and the properties of trigonometric functions in different quadrants is crucial for solving trigonometric equations.