Answer :
To determine the measure of angle BAC, we will follow these steps:
1. Understand the Problem:
- We have a right-angled triangle where the length of the opposite side to angle BAC is 3.1 units.
- The hypotenuse of the triangle is 4.5 units.
- We are asked to find the measure of angle BAC, denoted as [tex]\( x \)[/tex].
2. Recall the Trigonometric Function:
- The sine function relates the opposite side and the hypotenuse in a right-angled triangle.
- Specifically, [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
3. Set Up the Equation:
- For angle BAC, we have:
[tex]\[ \sin(x) = \frac{3.1}{4.5} \][/tex]
4. Solve for [tex]\( x \)[/tex] Using the Inverse Sine Function:
- [tex]\( \sin^{-1} \left( \frac{3.1}{4.5} \right) = x \)[/tex]
5. Calculate [tex]\( x \)[/tex]:
- We use the inverse sine (arcsine) function to find the angle.
- Using a calculator or trigonometric tables, find:
[tex]\[ x = \sin^{-1} \left( \frac{3.1}{4.5} \right) \][/tex]
6. Convert the Answer from Radians to Degrees (if necessary):
- Calculators typically provide the angle in degrees directly when finding the arcsine, but ensure you set your calculator to degree mode if needed.
7. Round to the Nearest Whole Degree:
- After calculating the angle, you get [tex]\( \approx 43.6 \)[/tex] degrees.
- Rounding this to the nearest whole number, we get [tex]\( 44 \)[/tex] degrees.
Thus, the measure of angle BAC is [tex]\( \boxed{44^\circ} \)[/tex].
1. Understand the Problem:
- We have a right-angled triangle where the length of the opposite side to angle BAC is 3.1 units.
- The hypotenuse of the triangle is 4.5 units.
- We are asked to find the measure of angle BAC, denoted as [tex]\( x \)[/tex].
2. Recall the Trigonometric Function:
- The sine function relates the opposite side and the hypotenuse in a right-angled triangle.
- Specifically, [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
3. Set Up the Equation:
- For angle BAC, we have:
[tex]\[ \sin(x) = \frac{3.1}{4.5} \][/tex]
4. Solve for [tex]\( x \)[/tex] Using the Inverse Sine Function:
- [tex]\( \sin^{-1} \left( \frac{3.1}{4.5} \right) = x \)[/tex]
5. Calculate [tex]\( x \)[/tex]:
- We use the inverse sine (arcsine) function to find the angle.
- Using a calculator or trigonometric tables, find:
[tex]\[ x = \sin^{-1} \left( \frac{3.1}{4.5} \right) \][/tex]
6. Convert the Answer from Radians to Degrees (if necessary):
- Calculators typically provide the angle in degrees directly when finding the arcsine, but ensure you set your calculator to degree mode if needed.
7. Round to the Nearest Whole Degree:
- After calculating the angle, you get [tex]\( \approx 43.6 \)[/tex] degrees.
- Rounding this to the nearest whole number, we get [tex]\( 44 \)[/tex] degrees.
Thus, the measure of angle BAC is [tex]\( \boxed{44^\circ} \)[/tex].
