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The problem provided asks to find the exact values of \(\sin \theta\) and \(\cos \theta\) given that \(\tan \theta = \frac{4}{3}\) and \(\csc \theta < 0\).

### Step-by-step solution:

1. **Tangent information:**
   \[
   \tan \theta = \frac{4}{3}
   \]
   This implies that the opposite side is 4 and the adjacent side is 3 in a right triangle.

2. **Pythagorean Theorem:**
   To find the hypotenuse (\(r\)), we apply the Pythagorean theorem:
   \[
   r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
   \]

3. **Sign determination from \(\csc \theta < 0\):**
   The condition \(\csc \theta < 0\) (which implies \(\sin \theta < 0\), since \(\csc \theta = \frac{1}{\sin \theta}\)) tells us that \(\theta\) must be in either the third or fourth quadrant, because sine is negative in those quadrants. However, because \(\tan \theta = \frac{4}{3}\) is positive, \(\theta\) must be in the third quadrant (where both sine and cosine are negative, and tangent is positive).

4. **Finding \(\sin \theta\) and \(\cos \theta\):**
   - In the third quadrant, both sine and cosine are negative.
   \[
   \sin \theta = -\frac{\text{opposite}}{\text{hypotenuse}} = -\frac{4}{5}
   \]
   \[
   \cos \theta = -\frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{3}{5}
   \]

Thus, the exact values are:
- \(\sin \theta = -\frac{4}{5}\)
- \(\cos \theta = -\frac{3}{5}\)

Would you like further clarification on any part of this? Here are 5 related questions you might explore:

1. How is the Pythagorean theorem applied to find the hypotenuse in a right triangle?
2. Why is \(\csc \theta < 0\) used to determine the quadrant of \(\theta\)?
3. How can you use the signs of sine and cosine to determine the quadrant of an angle?
4. What are the characteristics of trigonometric functions in different quadrants?
5. Can you derive \(\tan \theta\) from \(\sin \theta\) and \(\cos \theta\)?

**Tip:** Always check the quadrant when solving trigonometric problems to ensure the correct signs for \(\sin \theta\) and \(\cos \theta\).

Let theta be an angle such that tan(theta) = 4/3 and csc(theta) < 0. Find the exact values of sin(theta) and cos(theta).

Learn how to find the exact values of sin(theta) and cos(theta) for an angle where tan(theta) = 4/3 and csc(theta) is negative. This problem is solved using the Pythagorean Theorem and trigonometric ratios, suitable for high school students studying advanced trigonometry.

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