The value of `sin(2 sin^(-1)0.8)` is equal to

Options

A. `sin 1.2^@`
B. `sin1.6^@`
C. `0.48`
D. `0.96`

Answer

To solve the problem of finding the value of \( \sin(2 \sin^{-1}(0.8)) \), we can follow these steps:

### Step 1: Let \( y = \sin^{-1}(0.8) \)

This means that \( \sin(y) = 0.8 \).

### Step 2: Use the double angle formula for sine

We know that:
\[ \sin(2y) = 2 \sin(y) \cos(y) \]

### Step 3: Calculate \( \sin(y) \)

From Step 1, we have:
\[ \sin(y) = 0.8 \]

### Step 4: Calculate \( \cos(y) \)

Using the Pythagorean identity:
\[ \cos^2(y) + \sin^2(y) = 1 \]
we can find \( \cos(y) \):
\[ \cos^2(y) = 1 - \sin^2(y) = 1 - (0.8)^2 = 1 - 0.64 = 0.36 \]
Thus,
\[ \cos(y) = \sqrt{0.36} = 0.6 \]
(Note: Since \( y = \sin^{-1}(0.8) \) is in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), \( \cos(y) \) is positive.)

### Step 5: Substitute \( \sin(y) \) and \( \cos(y) \) into the double angle formula

Now we can substitute the values into the double angle formula:
\[ \sin(2y) = 2 \sin(y) \cos(y) = 2 \times 0.8 \times 0.6 \]

### Step 6: Calculate the final value

Calculating this gives:
\[ \sin(2y) = 2 \times 0.8 \times 0.6 = 2 \times 0.48 = 0.96 \]

Thus, the value of \( \sin(2 \sin^{-1}(0.8)) \) is \( 0.96 \).

### Final Answer:
\[ \sin(2 \sin^{-1}(0.8)) = 0.96 \]

---