To determine in which triangle the value of [tex]\( x \)[/tex] is equal to [tex]\( \tan^{-1}\left(\frac{3.1}{5.2}\right) \)[/tex], we need to consider the definition and properties of the inverse tangent (arctangent) function.
The expression [tex]\( \tan^{-1}\left(\frac{3.1}{5.2}\right) \)[/tex] represents an angle whose tangent (ratio of the opposite side to the adjacent side) is [tex]\(\frac{3.1}{5.2}\)[/tex].
Given that the arctangent function returns the angle [tex]\( x \)[/tex] for which [tex]\(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]:
1. Identify the triangle: We are looking for a right triangle where the lengths of the sides relative to the angle [tex]\( x \)[/tex] follow this ratio [tex]\( \frac{\text{opposite side}}{\text{adjacent side}} = \frac{3.1}{5.2} \)[/tex].
2. Opposite and Adjacent sides: In a right triangle, label the legs of the triangle such that one leg (opposite to angle [tex]\( x \)[/tex]) has length 3.1 units and the other leg (adjacent to angle [tex]\( x \)[/tex]) has length 5.2 units.
Thus, in the right triangle in question, for the angle [tex]\( x \)[/tex]:
- The side opposite the angle [tex]\( x \)[/tex] is 3.1 units.
- The side adjacent to the angle [tex]\( x \)[/tex] is 5.2 units.
The triangle in which the angle [tex]\( x \)[/tex] equal to [tex]\( \tan^{-1}\left(\frac{3.1}{5.2}\right) \)[/tex] must have these specific side lengths to satisfy the ratio [tex]\(\frac{3.1}{5.2}\)[/tex].
Therefore, the correct triangle is the one that has one leg of length 3.1 units and the other leg of length 5.2 units, forming a right angle between these two sides.
