https://file.aiquickdraw.com/mathbot/file/1728586448415b9gvtli0.jpg

In the given right triangle, \( \triangle ABC \), we know:

- Side \( AC \) and side \( BC \) are in the ratio \( AC : BC = 5 : 3 \).
- Angle \( a \) is opposite side \( BC \), making it the angle of interest.

### Step-by-step solution:

1. **Using trigonometric ratios:**
   Since it's a right triangle and we are given the ratio of the opposite side \( BC \) to the adjacent side \( AC \), we can use the tangent function:
   \[
   \tan(a) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{3}{5}
   \]
   
2. **Finding angle \( a \):**
   To find \( a \), we take the arctangent (inverse tangent) of \( \frac{3}{5} \):
   \[
   a = \tan^{-1}\left(\frac{3}{5}\right)
   \]
   Using a calculator, 
   \[
   a = \tan^{-1}(0.6) \approx 30.96^\circ
   \]

3. **Rounding to one decimal place:**
   \[
   a \approx 31.0^\circ
   \]

The calculated value of \( a \) is \( 31.0^\circ \), but in the image, it shows an answer of \( 60^\circ \). This suggests there may have been confusion about which angle was being solved for. If the problem asks for angle \( \angle A \), which is the complement of \( a \), then:
\[
\angle A = 90^\circ - a = 90^\circ - 31.0^\circ = 59.0^\circ
\]
Thus, rounding could give an answer close to \( 60^\circ \) as seen in the image.

Let me know if you'd like any clarifications!

### Related Questions:
1. What other trigonometric functions can be used to solve this problem?
2. How do we use the Pythagorean theorem to verify the sides of the triangle?
3. What would the sine and cosine of angle \( a \) be?
4. How can the angle \( A \) be calculated if both sides were known instead of a ratio?
5. How does changing the ratio of the sides affect the angle?

### Tip:
Always double-check whether you're asked to find an acute angle or its complement in a right triangle problem!

Given that Side AC : Side BC = 5 : 3, find the value of angle a.

Learn how to find an unknown angle in a right triangle using trigonometric ratios with a step-by-step solution. The problem involves a triangle with sides in a 5:3 ratio and requires calculating the angle using the tangent function. Suitable for Grades 8-10.

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