https://file.aiquickdraw.com/mathbot/file/17290919086047ealv28h.jpg

Let's analyze the right triangle in the image. The given triangle has:

- Hypotenuse = 5
- Adjacent side (to angle \(\theta\)) = 3
- Opposite side (to angle \(\theta\)) = 4 (based on the Pythagorean theorem: \(3^2 + 4^2 = 5^2\)).

Now, we can calculate the trigonometric values:

1. **\(\cot(\theta)\)** is the cotangent of the angle, which is the ratio of the adjacent side to the opposite side.
   \[
   \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{3}{4}
   \]

2. **\(\sin(\theta)\)** is the sine of the angle, which is the ratio of the opposite side to the hypotenuse.
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}
   \]

3. **\(\csc(\theta)\)** is the cosecant of the angle, which is the reciprocal of the sine.
   \[
   \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{4}{5}} = \frac{5}{4}
   \]

### Final answers:
- \(\cot(\theta) = \frac{3}{4}\)
- \(\sin(\theta) = \frac{4}{5}\)
- \(\csc(\theta) = \frac{5}{4}\)

Would you like more details or have any questions about this? Here are some related questions:

1. How does the Pythagorean theorem work in right triangles?
2. What is the difference between sine and cosecant functions?
3. Can you explain the relationship between the cotangent and tangent functions?
4. How do we derive trigonometric identities for other angles?
5. What are the common real-world applications of trigonometric ratios?

**Tip:** Memorizing the basic trigonometric ratios and their reciprocal relationships (like sine and cosecant) can make solving such problems faster!

Find cot(θ), sin(θ), and csc(θ), where θ is the angle shown in the figure. Give exact values, not decimal approximations.

Solve for cot(θ), sin(θ), and csc(θ) in a right triangle with sides 3, 4, and 5. Learn how to apply trigonometric ratios and the Pythagorean Theorem. Ideal for students in Grades 9-12.

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