To find the measure of the unknown angle [tex]\( x \)[/tex] that satisfies the equation [tex]\( x = \sin^{-1}\left(\frac{5}{8.3}\right) \)[/tex], we need to follow a systematic approach.
### Step-by-Step Solution:
1. Understand the Problem:
We are given the expression [tex]\( \sin^{-1}\left(\frac{5}{8.3}\right) \)[/tex] and need to determine the angle [tex]\( x \)[/tex] satisfying this condition.
2. Calculate the Sine Value:
Here, [tex]\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)[/tex]. Thus, the value given inside the inverse sine function is [tex]\(\frac{5}{8.3}\)[/tex].
3. Determine the Angle in Radians:
The inverse sine function will provide the angle in radians. Given:
[tex]\[
\theta = \sin^{-1}\left(\frac{5}{8.3}\right) \approx 0.6465165714340122 \text{ radians}
\][/tex]
4. Convert Radians to Degrees:
Since the angle is often more interpretable in degrees, we can convert the result from radians to degrees using the conversion factor [tex]\( 180^\circ / \pi \)[/tex]. The calculated angle is equivalent to:
[tex]\[
\theta \approx 37.0426709284371^\circ
\][/tex]
Thus, the measure of the unknown angle [tex]\( x \)[/tex] in a triangle where [tex]\(\sin^{-1}\left(\frac{5}{8.3}\right)\)[/tex] holds is approximately:
- [tex]\(\theta \approx 0.6465\)[/tex] radians, or
- [tex]\(\theta \approx 37.0427\)[/tex] degrees.
Therefore, the unknown angle [tex]\( x \)[/tex] is approximately [tex]\( 0.6465 \)[/tex] radians or [tex]\( 37.0427 \)[/tex] degrees. This completes our detailed, step-by-step solution.
