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The image shows a right triangle \( \triangle ABC \), where \( AB = 8 \), \( BC = 5 \), and angle \( \angle C = 90^\circ \). The task is to find trigonometric ratios for the given triangle, specifically the values of sine, cosine, and tangent for angles \( A \) and \( B \).

### Step 1: Find the Hypotenuse
To find the hypotenuse \( AC \), we can use the Pythagorean theorem:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89}.
\]
Thus, \( AC = \sqrt{89} \).

### Step 2: Trigonometric Ratios

1. **For angle \( A \)**:
   - \( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{5}{\sqrt{89}} \).
   - \( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{8}{\sqrt{89}} \).
   - \( \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{5}{8} \).

2. **For angle \( B \)**:
   - \( \sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{8}{\sqrt{89}} \).
   - \( \cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{5}{\sqrt{89}} \).
   - \( \tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{8}{5} \).

### Summary of Results:
1. \( \sin(A) = \frac{5}{\sqrt{89}} \).
2. \( \cos(A) = \frac{8}{\sqrt{89}} \).
3. \( \tan(A) = \frac{5}{8} \).
4. \( \sin(B) = \frac{8}{\sqrt{89}} \).
5. \( \cos(B) = \frac{5}{\sqrt{89}} \).
6. \( \tan(B) = \frac{8}{5} \).

Would you like me to simplify these ratios or provide more details?

Here are five related questions you might find helpful:
1. How do we rationalize the denominator in trigonometric ratios?
2. How can we use these trigonometric ratios to find the angles \( A \) and \( B \)?
3. How does the Pythagorean identity relate to these trigonometric functions?
4. What are the reciprocal trigonometric functions for angles \( A \) and \( B \)?
5. How do trigonometric ratios change as the angle increases or decreases in a right triangle?

**Tip:** When finding trigonometric ratios, always double-check that the correct sides are used for each ratio relative to the angle.

In the following right triangle, find sin A, cos A, tan A, sin B, cos B, and tan B, where AB = 8, BC = 5, and angle C = 90 degrees.

Learn how to find the trigonometric ratios (sin, cos, tan) for a right triangle where AB = 8, BC = 5, and angle C = 90°. Step-by-step application of the Pythagorean theorem and trigonometric formulas is used. Suitable for students in Grades 9-11.

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