Sure! Let's solve the problem step-by-step.
### Problem: Determine \( x \) in a right triangle given the length of the opposite side is \( 3.1 \) units and the length of the adjacent side is \( 5.2 \) units. The value of \( x \) is given as \( \tan^{-1}\left(\frac{3.1}{5.2}\right) \).
### Step-by-Step Solution:
1. Understand 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, \(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\).
2. Given Values:
- Opposite side = \( 3.1 \) units
- Adjacent side = \( 5.2 \) units
3. Find the Tangent Value:
Calculate the ratio of the opposite side to the adjacent side:
[tex]\[
\frac{\text{opposite}}{\text{adjacent}} = \frac{3.1}{5.2}
\][/tex]
4. Use the Inverse Tangent Function:
The angle \( x \) can be found by taking the inverse tangent (\(\tan^{-1}\)) of the ratio calculated above. So,
[tex]\[
x = \tan^{-1}\left(\frac{3.1}{5.2}\right)
\][/tex]
5. Calculate the Angle:
Using a calculator or mathematical software, we can find the value of \( x \). The value of \( x \) in radians is approximately:
[tex]\[
x \approx 0.5376 \text{ radians}
\][/tex]
6. Convert Radians to Degrees:
To convert the angle from radians to degrees, we use the conversion factor \( 180^\circ / \pi \). So,
[tex]\[
x \text{ (in degrees)} \approx 0.5376 \times \frac{180^\circ}{\pi} \approx 30.8014^\circ
\][/tex]
### Conclusion:
The angle [tex]\( x \)[/tex] in the given right triangle, where the opposite side is [tex]\( 3.1 \)[/tex] units and the adjacent side is [tex]\( 5.2 \)[/tex] units, is approximately [tex]\( 0.5376 \)[/tex] radians or [tex]\( 30.8014^\circ \)[/tex].
