`sin(cos^(-1)(3/5))` is equal to

Options

A. `pi/2-3/5`
B. `3/5`
C. `-4/5`
D. `4/5`

Answer

To solve the problem \( \sin(\cos^{-1}(\frac{3}{5})) \), we will follow these steps:

### Step 1: Set up the equation
Let \( x = \cos^{-1}(\frac{3}{5}) \). This implies that \( \cos(x) = \frac{3}{5} \).

### Step 2: Draw a right triangle
To visualize this, we can draw a right triangle where:
- The angle \( x \) is one of the angles in the triangle.
- The adjacent side (base) to angle \( x \) is 3.
- The hypotenuse is 5.

### Step 3: Use the Pythagorean theorem to find the opposite side
According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
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 \( \sin(x) \)
Now that we have the lengths of all sides of the triangle, we can find \( \sin(x) \):
\[ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]

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

### Final Answer
The final answer is:
\[ \frac{4}{5} \]

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