In a right ΔABC, right angled at C, AC = 3cm, AB = 5cm. Find the value of (i) sin A, cos A (ii) sin B, cos B

Given the right triangle \( \Delta ABC \) with a right angle at \( C \), \( AC = 3 \, \text{cm} \), and \( AB = 5 \, \text{cm} \), we are to find the values of \( \sin A \), \( \cos A \), \( \sin B \), and \( \cos B \).

### Step 1: Determine the length of \( BC \) using the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle:
\[
AB^2 = AC^2 + BC^2
\]
Substituting the known values:
\[
5^2 = 3^2 + BC^2
\]
\[
25 = 9 + BC^2
\]
\[
BC^2 = 25 - 9 = 16
\]
\[
BC = \sqrt{16} = 4 \, \text{cm}
\]

Now, the sides of the triangle are:
- \( AC = 3 \, \text{cm} \)
- \( BC = 4 \, \text{cm} \)
- \( AB = 5 \, \text{cm} \) (the hypotenuse)

### Step 2: Calculate the trigonometric ratios

#### (i) For angle \( A \):
- \( \sin A = \frac{\text{Opposite to } A}{\text{Hypotenuse}} = \frac{BC}{AB} = \frac{4}{5} \)
- \( \cos A = \frac{\text{Adjacent to } A}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{3}{5} \)

So:
\[
\sin A = \frac{4}{5}, \quad \cos A = \frac{3}{5}
\]

#### (ii) For angle \( B \):
- \( \sin B = \frac{\text{Opposite to } B}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{3}{5} \)
- \( \cos B = \frac{\text{Adjacent to } B}{\text{Hypotenuse}} = \frac{BC}{AB} = \frac{4}{5} \)

So:
\[
\sin B = \frac{3}{5}, \quad \cos B = \frac{4}{5}
\]

### Summary:
- \( \sin A = \frac{4}{5}, \quad \cos A = \frac{3}{5} \)
- \( \sin B = \frac{3}{5}, \quad \cos B = \frac{4}{5} \)

Would you like any further details or have any questions?

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Here are 5 related questions:
1. What is the value of \( \tan A \) and \( \tan B \) in \( \Delta ABC \)?
2. How does the sine and cosine of angles relate to each other in a right triangle?
3. If the angle \( A \) increases, how do \( \sin A \) and \( \cos A \) change?
4. How would you find the area of \( \Delta ABC \) using trigonometric ratios?
5. Can you prove the Pythagorean theorem using trigonometric identities?

**Tip:** The sine and cosine of complementary angles in a right triangle are equal; i.e., \( \sin A = \cos B \) and \( \cos A = \sin B \).

Explore how to find the sine and cosine values of angles A and B in a right triangle with sides AC = 3cm, AB = 5cm. This problem uses the Pythagorean Theorem and trigonometric ratios to determine sin A, cos A, sin B, and cos B. Suitable for Grades 9-10.

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