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To solve this, let's analyze the triangle and determine the trigonometric ratios of angle \( C \).

### Given Information:
- \( \triangle ABC \) is a right-angled triangle at \( B \).
- \( 5 \cdot AB = 3 \cdot AC \).

Since \( \triangle ABC \) is right-angled at \( B \), we can label the sides:
- \( AB \): the leg opposite angle \( C \),
- \( BC \): the leg adjacent to angle \( C \),
- \( AC \): the hypotenuse.

### Step 1: Using the Ratio
We are given:
\[
5 \cdot AB = 3 \cdot AC
\]
Let's assume:
\[
AC = 5x \quad \text{and} \quad AB = 3x
\]
for some positive constant \( x \). 

### Step 2: Applying the Pythagorean Theorem
In \( \triangle ABC \),
\[
AC^2 = AB^2 + BC^2
\]
Substituting \( AB = 3x \) and \( AC = 5x \),
\[
(5x)^2 = (3x)^2 + BC^2
\]
\[
25x^2 = 9x^2 + BC^2
\]
\[
BC^2 = 16x^2
\]
\[
BC = 4x
\]

### Step 3: Trigonometric Ratios for Angle \( C \)
Now we have:
- \( AB = 3x \),
- \( BC = 4x \),
- \( AC = 5x \).

The trigonometric ratios for angle \( C \) are:
1. **Sine** \( \sin C = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{3x}{5x} = \frac{3}{5} \).
2. **Cosine** \( \cos C = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{4x}{5x} = \frac{4}{5} \).
3. **Tangent** \( \tan C = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{3x}{4x} = \frac{3}{4} \).

### Answer
The main trigonometric ratios for angle \( C \) are:
- \( \sin C = \frac{3}{5} \)
- \( \cos C = \frac{4}{5} \)
- \( \tan C = \frac{3}{4} \)

Would you like a deeper breakdown of any step, or do you have further questions?

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Here are some related questions for practice:
1. How would the ratios change if the given condition was \( 4AB = 3AC \)?
2. What are the trigonometric ratios of angle \( A \) in this triangle?
3. How does the Pythagorean Theorem verify the lengths of sides in a right triangle?
4. If \( AB \) is increased, how does it affect the value of \( \sin C \)?
5. Can you find \( \cot C \) based on the calculated values?

**Tip:** Remember that trigonometric ratios are defined based on the relationship between specific sides and the angle in question.

ABC is a right-angled triangle at B. If 5AB = 3AC, find the main trigonometric ratios of angle C.

Discover the solution for finding the main trigonometric ratios of angle C in a right-angled triangle ABC at B with the condition 5AB = 3AC. This problem uses trigonometry concepts, including sine, cosine, tangent, and the Pythagorean theorem, perfect for Grades 9-10.

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