Given a right triangle with an angle $x$, in which triangle is the value of $x$ equal to ${\tan }^{-1}\left(\frac{3.1}{5.2}\right)$?

• A. A triangle where the side opposite angle $x$ is $5.2$ units and the hypotenuse is $3.1$ units.
• B. A triangle where the hypotenuse is $3.1$ units and the side adjacent to angle $x$ is $5.2$ units.
• C. A triangle where the side opposite angle $x$ is $5.2$ units and the side adjacent to angle $x$ is $3.1$ units.
• D. A triangle where the side opposite angle $x$ is $3.1$ units and the side adjacent to angle $x$ is $5.2$ units.

Answer: (D)

Verified step by step guidance

1. Recall the definition of the tangent function in a right triangle: \(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\).
2. Given that \(x = \tan^{-1}\left(\frac{3.1}{5.2}\right)\), this means the ratio of the side opposite angle \(x\) to the side adjacent to angle \(x\) is \(\frac{3.1}{5.2}\).
3. Identify which sides correspond to the opposite and adjacent sides relative to angle \(x\) in the triangle options provided.
4. Check each triangle option to see if the ratio \(\frac{\text{opposite}}{\text{adjacent}}\) equals \(\frac{3.1}{5.2}\).
5. Select the triangle where the side opposite angle \(x\) is 3.1 units and the side adjacent to angle \(x\) is 5.2 units, as this matches the given tangent inverse expression.