To determine the measure of the unknown angle [tex]\( x \)[/tex] in the triangle, given that [tex]\( x = \sin^{-1}\left(\frac{5}{8.3}\right) \)[/tex], we'll walk through the steps needed to solve for [tex]\( x \)[/tex].
### Step 1: Understand the Problem
The equation [tex]\( x = \sin^{-1}\left(\frac{5}{8.3}\right) \)[/tex] is asking us to find the angle [tex]\( x \)[/tex] whose sine value is [tex]\( \frac{5}{8.3} \)[/tex]. Here, [tex]\(\sin^{-1}\)[/tex] denotes the inverse sine function, also known as arcsine.
### Step 2: Calculate the Sine Value
First, we'll interpret what [tex]\(\sin^{-1}(y)\)[/tex] means. It gives us the angle whose sine is [tex]\( y \)[/tex].
Given our problem:
[tex]\[ \sin(x) = \frac{5}{8.3} \][/tex]
### Step 3: Find the Angle [tex]\( x \)[/tex] in Radians
Using the arcsine function:
[tex]\[ x = \sin^{-1}\left(\frac{5}{8.3}\right) \][/tex]
From a correctly executed calculation, the angle [tex]\( x \)[/tex] in radians is:
[tex]\[ x \approx 0.6465 \text{ radians} \][/tex]
### Step 4: Convert to Degrees (if required)
Oftentimes, angles in trigonometry are expressed in degrees rather than radians. To convert the radians result to degrees, we use the conversion factor:
[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]
So:
[tex]\[ x \approx 0.6465 \text{ radians} \times \left(\frac{180}{\pi}\right) \approx 37.0427 \text{ degrees} \][/tex]
### Step 5: State the Result
Hence, the measure of the unknown angle [tex]\( x \)[/tex] is approximately:
[tex]\[ x \approx 0.6465 \text{ radians} \][/tex]
or equivalently:
[tex]\[ x \approx 37.04^\circ \][/tex]
### Conclusion
In the given triangle, the angle [tex]\( x \)[/tex] corresponding to [tex]\(\sin^{-1}\left(\frac{5}{8.3}\right)\)[/tex] is measured as approximately 0.6465 radians or 37.04 degrees.
