To determine which triangle corresponds to the value [tex]\( x = \tan^{-1}\left(\frac{3.1}{5.2}\right) \)[/tex], we need to recall the relationship between the tangent function and the sides of a right triangle.
1. Understanding the Tangent Function:
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically,
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
2. Given Information:
We are given that:
[tex]\[
x = \tan^{-1}\left(\frac{3.1}{5.2}\right)
\][/tex]
This implies the tangent of angle [tex]\( x \)[/tex] is [tex]\( \frac{3.1}{5.2} \)[/tex].
3. Interpreting the Tangent Ratio:
[tex]\[
\frac{3.1}{5.2}
\][/tex]
In the context of a right triangle, this means:
- The length of the side opposite to angle [tex]\( x \)[/tex] is 3.1 units.
- The length of the side adjacent to angle [tex]\( x \)[/tex] is 5.2 units.
4. Identifying the Triangle:
To determine which triangle corresponds to this information, look for a right triangle with:
- One side measuring 3.1 units
- An adjacent side measuring 5.2 units
5. Conclusion:
The triangle in which the value of [tex]\( x \)[/tex] is [tex]\(\tan^{-1}\left(\frac{3.1}{5.2}\right)\)[/tex] will have a right angle, one side (opposite to [tex]\( x \)[/tex]) of length 3.1 units, and another side (adjacent to [tex]\( x \)[/tex]) of length 5.2 units.
Based on the tangent calculation, the correct triangle has these specific side lengths. The measure of [tex]\( x \)[/tex] in this triangle will be approximately 0.5376 radians.
