To determine the measure of the unknown angle [tex]\( x \)[/tex] in a right triangle where the opposite side is 5 and the hypotenuse is 8.3, follow these steps:
1. Understand the problem: We are given the lengths of the opposite side and the hypotenuse of a right triangle, and we need to find the measure of the angle [tex]\( x \)[/tex] using the inverse sine function ([tex]\(\sin^{-1}\)[/tex]).
2. Set up the ratio: The sine of angle [tex]\( x \)[/tex] is defined as the ratio of the length of the opposite side to the hypotenuse. Therefore, we have:
[tex]\[
\sin(x) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{5}{8.3}
\][/tex]
3. Calculate the angle: To find the angle [tex]\( x \)[/tex], we need to take the inverse sine ([tex]\(\sin^{-1}\)[/tex]) of the ratio [tex]\(\frac{5}{8.3}\)[/tex]:
[tex]\[
x = \sin^{-1}\left(\frac{5}{8.3}\right)
\][/tex]
This involves using a scientific calculator or an appropriate mathematical tool to find the angle in radians and then converting it to degrees if necessary.
4. Determine the value: By evaluating the inverse sine, we find:
[tex]\[
x \approx 0.6465165714340122 \text{ radians}
\][/tex]
5. Convert radians to degrees (if needed): To convert from radians to degrees, use the conversion factor [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[
x \times \frac{180}{\pi} \approx 0.6465165714340122 \times 57.2958 \approx 37.0426709284371^\circ
\][/tex]
Therefore, the measure of the unknown angle [tex]\( x \)[/tex] is approximately 0.6465 radians or 37.04 degrees.
