If: `tan^(-1)x = sin^(-1)(3/sqrt(10))`, then: x=

Options

A. 3
B. `-3`
C. `1/3`
D. `-1/3`

Answer

To solve the equation \( \tan^{-1} x = \sin^{-1} \left( \frac{3}{\sqrt{10}} \right) \), we can follow these steps:

### Step 1: Set \( \tan^{-1} x \) as \( \theta \)
Let \( \theta = \tan^{-1} x \). This implies that \( x = \tan \theta \).

**Hint:** Remember that the tangent function relates the opposite side to the adjacent side in a right triangle.

### Step 2: Relate \( \sin^{-1} \left( \frac{3}{\sqrt{10}} \right) \) to \( \theta \)
From the equation, we have:
\[ \theta = \sin^{-1} \left( \frac{3}{\sqrt{10}} \right) \]

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

### Step 3: Construct a right triangle
In the triangle corresponding to \( \theta \):
- The opposite side = 3
- The hypotenuse = \( \sqrt{10} \)

Using the Pythagorean theorem, we can find the adjacent side:
\[ \text{Adjacent} = \sqrt{\text{Hypotenuse}^2 - \text{Opposite}^2} = \sqrt{10 - 9} = 1 \]

**Hint:** Use the Pythagorean theorem \( a^2 + b^2 = c^2 \) to find the missing side.

### Step 4: Find \( \tan \theta \)
Now we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{1} = 3 \]

**Hint:** Recall that tangent is the ratio of the opposite side to the adjacent side.

### Step 5: Substitute back to find \( x \)
Since \( x = \tan \theta \), we have:
\[ x = 3 \]

**Hint:** Ensure that you substitute back correctly to find the value of \( x \).

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{3} \]