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To find the exact values of the six trigonometric ratios (sin, cos, tan, csc, sec, cot) for the angle \(\theta\) in the given right triangle, we need to use the given side lengths and the Pythagorean theorem.

### Step 1: Identify the sides of the triangle
- **Opposite side** to angle \(\theta\): \(60\)
- **Adjacent side** to angle \(\theta\): \(32\)
- **Hypotenuse**: To be determined.

### Step 2: Calculate the hypotenuse
Using the Pythagorean theorem:
\[
\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2
\]
\[
\text{Hypotenuse}^2 = 60^2 + 32^2
\]
\[
\text{Hypotenuse}^2 = 3600 + 1024 = 4624
\]
\[
\text{Hypotenuse} = \sqrt{4624} = 68
\]

### Step 3: Find the trigonometric ratios

1. **Sine (sin) of \(\theta\):**
   \[
   \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{60}{68} = \frac{15}{17}
   \]

2. **Cosine (cos) of \(\theta\):**
   \[
   \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{32}{68} = \frac{8}{17}
   \]

3. **Tangent (tan) of \(\theta\):**
   \[
   \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{60}{32} = \frac{15}{8}
   \]

4. **Cosecant (csc) of \(\theta\):**
   \[
   \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{17}{15}
   \]

5. **Secant (sec) of \(\theta\):**
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{17}{8}
   \]

6. **Cotangent (cot) of \(\theta\):**
   \[
   \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{8}{15}
   \]

### Summary of Trigonometric Ratios

- \(\sin(\theta) = \frac{15}{17}\)
- \(\cos(\theta) = \frac{8}{17}\)
- \(\tan(\theta) = \frac{15}{8}\)
- \(\csc(\theta) = \frac{17}{15}\)
- \(\sec(\theta) = \frac{17}{8}\)
- \(\cot(\theta) = \frac{8}{15}\)

Would you like more details or have any questions? 

### 5 Related Questions:
1. How do you derive the Pythagorean theorem?
2. What are the definitions of the six trigonometric functions?
3. How can these trigonometric ratios be used in real-life applications?
4. How do we find trigonometric ratios for angles other than those in right triangles?
5. What is the significance of trigonometric identities in simplifying expressions?

### Tip:
Always keep in mind that the hypotenuse is always the longest side of a right triangle. This can help you quickly verify your calculations!

Find the exact values of the six trigonometric ratios of the angle θ in the triangle.

Learn how to calculate the exact values of the six trigonometric ratios for the angle θ in a right triangle with side lengths of 60 and 32. This problem involves applying the Pythagorean theorem and basic trigonometric formulas, suitable for high school students.

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