Evaluate (i) `sin (cos^(-1) (3/5))`

Answer

To evaluate \( \sin(\cos^{-1}(3/5)) \), we can follow these steps:

### Step 1: Let \( \theta = \cos^{-1}(3/5) \)
This means that \( \cos(\theta) = \frac{3}{5} \).

**Hint:** Remember that \( \cos^{-1}(x) \) gives you an angle whose cosine is \( x \).

### Step 2: Draw a right triangle
In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side (base) to the hypotenuse. Here, we can represent this as:
- Adjacent side (base) = 3
- Hypotenuse = 5

**Hint:** Visualizing the problem with a triangle can help you understand the relationships between the sides and angles.

### Step 3: Find the length of the opposite side (perpendicular)
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + \text{Perpendicular}^2 \]
\[ 25 = 9 + \text{Perpendicular}^2 \]
\[ \text{Perpendicular}^2 = 25 - 9 = 16 \]
\[ \text{Perpendicular} = \sqrt{16} = 4 \]

**Hint:** The Pythagorean theorem is a key tool for finding missing side lengths in right triangles.

### Step 4: Calculate \( \sin(\theta) \)
Now that we have the lengths of all sides, we can find \( \sin(\theta) \):
\[ \sin(\theta) = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{4}{5} \]

**Hint:** The sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle.

### Step 5: Conclusion
Thus, we have:
\[ \sin(\cos^{-1}(3/5)) = \frac{4}{5} \]

**Final Answer:** \( \frac{4}{5} \)