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 understand the relationship given by the arctangent function.
### Step-by-Step Solution:
1. Understand the Problem:
- We're given an angle [tex]\( x \)[/tex] defined as [tex]\( \tan^{-1} \left( \frac{3.1}{5.2} \right) \)[/tex].
- This means that in a right triangle, [tex]\( x \)[/tex] is the angle whose tangent is the ratio of the lengths of the opposite side to the adjacent side, which is [tex]\( \frac{3.1}{5.2} \)[/tex].
2. Calculate the Angle:
- The angle [tex]\( x \)[/tex] can be computed using the arctangent function, which retrieves the angle for a given tangent value.
- In this specific case, [tex]\( x \approx 0.5376 \)[/tex] radians.
- Converting this angle from radians to degrees (as typically angles in geometry are in degrees), we get [tex]\( x \approx 30.80^\circ \)[/tex].
3. Relate to Triangle Properties:
- For an angle [tex]\( x \)[/tex] in a right triangle:
[tex]\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Given:
[tex]\[
\tan(x) = \frac{3.1}{5.2}
\][/tex]
4. Check the Given Options:
- Each triangle will have sides labeled with lengths corresponding to the opposite and adjacent sides from the angle [tex]\( x \)[/tex].
- To find the correct triangle, we simply find one where the ratio of the lengths of these sides matches [tex]\( \frac{3.1}{5.2} \)[/tex].
5. Conclusion:
- The correct triangle has an angle [tex]\( x \)[/tex] such that the sides opposite and adjacent to [tex]\( x \)[/tex] have lengths 3.1 and 5.2, respectively.
- Therefore, the angle [tex]\( x \)[/tex] in this triangle will be approximately [tex]\( 30.80^\circ \)[/tex].
### Final Answer:
The correct triangle is the one where the ratio of the length of the opposite side to the adjacent side to angle [tex]\( x \)[/tex] is [tex]\( \frac{3.1}{5.2} \)[/tex]. This corresponds to the angle [tex]\( x \approx 30.80^\circ \)[/tex].
