Let empty set be an angle in quadrant 111 such that cos empty set = -3/5 .
Find the exact values of csc empty set and tan empty set.

Given that \(\cos \theta = -\frac{3}{5}\) and \(\theta\) is an angle in the third quadrant, we need to find the exact values of \(\csc \theta\) and \(\tan \theta\).

### Step-by-Step Solution

1. **Determine \(\sin \theta\)**:
   Since \(\theta\) is in the third quadrant, both \(\sin \theta\) and \(\cos \theta\) are negative. We can use the Pythagorean identity:
   \[
   \sin^2 \theta + \cos^2 \theta = 1
   \]
   Substituting \(\cos \theta = -\frac{3}{5}\):
   \[
   \sin^2 \theta + \left( -\frac{3}{5} \right)^2 = 1
   \]
   \[
   \sin^2 \theta + \frac{9}{25} = 1
   \]
   \[
   \sin^2 \theta = 1 - \frac{9}{25}
   \]
   \[
   \sin^2 \theta = \frac{25}{25} - \frac{9}{25}
   \]
   \[
   \sin^2 \theta = \frac{16}{25}
   \]
   Taking the square root and considering the sign of \(\sin \theta\) in the third quadrant:
   \[
   \sin \theta = -\frac{4}{5}
   \]

2. **Determine \(\csc \theta\)**:
   \(\csc \theta\) is the reciprocal of \(\sin \theta\):
   \[
   \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}
   \]

3. **Determine \(\tan \theta\)**:
   \(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\):
   \[
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}
   \]

### Exact Values
- \(\csc \theta = -\frac{5}{4}\)
- \(\tan \theta = \frac{4}{3}\)

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:
1. How do you find the exact values of the other trigonometric functions given one trigonometric ratio?
2. Can you explain the Pythagorean identity in trigonometry?
3. How do the signs of trigonometric functions change in different quadrants?
4. What is the relationship between \(\sec \theta\) and \(\cos \theta\)?
5. How do you solve trigonometric equations involving multiple angles?

**Tip:** When working with trigonometric functions, always consider the quadrant in which the angle lies, as this determines the signs of the functions.

Explore how to find the exact values of cosecant and tangent for an angle \(\theta\) in the third quadrant with a cosine of -3/5. Detailed step-by-step solution provided.

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