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

Answer

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

### Step 1: Understand the expression
We have \( \cos^{-1}(\frac{3}{5}) \). This means we are looking for an angle \( \theta \) such that \( \cos(\theta) = \frac{3}{5} \).

### Step 2: Draw a right triangle
To visualize this, we can draw a right triangle where:
- The adjacent side (base) is 3,
- The hypotenuse is 5.

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

### Step 4: Calculate the sine of the angle
Now we can find \( \sin(\theta) \) where \( \theta = \cos^{-1}(\frac{3}{5}) \):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]

### Step 5: Conclusion
Thus, the value of \( \sin[\cos^{-1}(\frac{3}{5})] \) is:
\[ \sin[\cos^{-1}(\frac{3}{5})] = \frac{4}{5} \]

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