To determine the measure of angle [tex]\( \angle BAC \)[/tex], we can use trigonometry, specifically the sine function. Here's the step-by-step process:
1. Identify the given quantities:
- Opposite side ([tex]\( \text{opposite} \)[/tex]) = 3.1 units
- Hypotenuse ([tex]\( \text{hypotenuse} \)[/tex]) = 4.5 units
2. Set up the equation:
- The given equation to find the angle [tex]\( x \)[/tex] is [tex]\( \sin^{-1}\left(\frac{3.1}{4.5}\right) = x \)[/tex].
3. Compute the ratio:
- Calculate the ratio of the opposite side to the hypotenuse:
[tex]\[
\frac{3.1}{4.5} \approx 0.68888889
\][/tex]
4. Apply the arcsine function:
- Find the angle [tex]\( x \)[/tex] whose sine is the ratio calculated:
[tex]\[
x = \sin^{-1}(0.68888889)
\][/tex]
- This gives us an angle in radians. Let's denote this angle in radians as [tex]\( \text{angle\_rad} \)[/tex].
5. Convert radians to degrees:
- Convert [tex]\( \text{angle\_rad} \)[/tex] to degrees (since the question asks for the measure in degrees):
[tex]\[
\text{angle\_deg} = \text{angle\_rad} \times \frac{180}{\pi}
\][/tex]
- The exact value of [tex]\( \text{angle\_deg} \)[/tex] is approximately 43.542 degrees.
6. Round to the nearest whole degree:
- Rounding [tex]\( 43.542 \)[/tex] degrees to the nearest whole number gives us [tex]\( 44 \)[/tex] degrees.
Therefore, the measure of [tex]\( \angle BAC \)[/tex] is [tex]\( \boxed{44 \text{ degrees}} \)[/tex].
