If: `sin^(-1)(1/2)= tan^(-1)x`, then x=

Options

A. `sqrt(3)`
B. `1//sqrt(3)`
C. `1//sqrt(2)`
D. none of these

Answer

To solve the equation \( \sin^{-1}\left(\frac{1}{2}\right) = \tan^{-1}(x) \), we will follow these steps:

### Step 1: Find the value of \( \sin^{-1}\left(\frac{1}{2}\right) \)

We know that \( \sin(\theta) = \frac{1}{2} \) corresponds to a specific angle. The angle whose sine is \( \frac{1}{2} \) is \( 30^\circ \) or \( \frac{\pi}{6} \) radians.

\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Step 2: Set up the equation

Now we can substitute this value back into our original equation:

\[ \frac{\pi}{6} = \tan^{-1}(x) \]

### Step 3: Take the tangent of both sides

To eliminate the inverse tangent, we take the tangent of both sides:

\[ \tan\left(\frac{\pi}{6}\right) = x \]

### Step 4: Calculate \( \tan\left(\frac{\pi}{6}\right) \)

We know from trigonometric values that:

\[ \tan\left(\frac{\pi}{6}\right) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 5: Conclude the value of \( x \)

Thus, we find that:

\[ x = \frac{1}{\sqrt{3}} \]

### Final Answer

The value of \( x \) is \( \frac{1}{\sqrt{3}} \).

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