To determine the value of [tex]\( x \)[/tex] in a right triangle where the opposite side is 3.1 and the adjacent side is 5.2, we follow these steps:
1. Identify the trigonometric ratio involved: Since we have the lengths of both the opposite side (3.1) and the adjacent side (5.2) of a right triangle, we use the tangent function, which is defined as the ratio of the opposite side to the adjacent side. Thus, [tex]\( x = \tan^{-1} \left( \frac{\text{opposite}}{\text{adjacent}} \right) \)[/tex].
2. Set up the equation: Substituting the given values,
[tex]\[
x = \tan^{-1} \left( \frac{3.1}{5.2} \right)
\][/tex]
3. Calculate the fraction: Compute the value inside the arctangent function,
[tex]\[
\frac{3.1}{5.2} \approx 0.5961538461538461
\][/tex]
4. Find the inverse tangent (arctan) of the result:
[tex]\[
x = \tan^{-1} (0.5961538461538461)
\][/tex]
The approximate value of [tex]\( x \)[/tex] in radians is:
[tex]\[
x \approx 0.5376 \, \text{radians}
\][/tex]
5. Convert radians to degrees (optional): Since most people find degrees easier to understand, we convert the angle from radians to degrees. The conversion factor from radians to degrees is [tex]\(\frac{180}{\pi}\)[/tex].
[tex]\[
x \, \text{(in degrees)} = 0.5376 \times \frac{180}{\pi} \approx 30.80^\circ
\][/tex]
Therefore, in the right triangle where the opposite side is 3.1 and the adjacent side is 5.2, the value of the angle [tex]\( x \)[/tex] is approximately [tex]\( 0.5376 \)[/tex] radians or [tex]\( 30.80^\circ \)[/tex].
