To find the measure of the unknown angle [tex]\( x \)[/tex] in a triangle where [tex]\(\sin^{-1}\left(\frac{5}{8.3}\right)\)[/tex], we need to follow a series of logical steps.
1. Identify the given information:
- We know the length of the opposite side of an angle, [tex]\(x\)[/tex], is 5 units.
- We also know the length of the hypotenuse of the triangle is 8.3 units.
2. Apply the inverse sine function:
- The function [tex]\(\sin^{-1}\)[/tex] (or arcsine) is used to determine the angle when the ratio of the opposite side to the hypotenuse is known.
- We use the ratio of the opposite side to the hypotenuse, which is [tex]\(\frac{5}{8.3}\)[/tex].
3. Calculate the angle in radians:
- The [tex]\( \sin^{-1} \left(\frac{5}{8.3}\right) \)[/tex] gives the angle in radians.
- According to the result, the angle in radians is: [tex]\(0.6465165714340122\)[/tex].
4. Convert the angle to degrees:
- Angles measured in radians can be converted to degrees using the conversion factor [tex]\(180^\circ / \pi \)[/tex], where [tex]\(\pi\)[/tex] is a mathematical constant approximately equal to 3.14159.
- The given result for the angle in degrees is: [tex]\(37.0426709284371^\circ\)[/tex].
Therefore, the measure of the unknown angle [tex]\(x\)[/tex] is approximately [tex]\(37.0426709284371^\circ\)[/tex].
