To find the area of the given right triangle, we can follow these steps:
1. Identify the given values:
- Adjacent leg ([tex]\(a\)[/tex]) = [tex]\(27.6 \, cm\)[/tex]
- Hypotenuse ([tex]\(c\)[/tex]) = [tex]\(30 \, cm\)[/tex]
- Angle between the adjacent leg and the hypotenuse ([tex]\(\theta\)[/tex]) = [tex]\(23^\circ\)[/tex]
2. Calculate the angle in radians:
Since angles in trigonometric functions are usually considered in radians, convert [tex]\(23^\circ\)[/tex] to radians:
[tex]\[
\theta = 23^\circ \approx 0.4027 \, \text{radians} \quad (\text{as found in the answer})
\][/tex]
3. Find the length of the opposite leg:
Use the sine function which relates the opposite leg ([tex]\(b\)[/tex]) to the hypotenuse ([tex]\(c\)[/tex]):
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
[tex]\[
\sin(23^\circ) = \frac{b}{30 \, cm}
\][/tex]
[tex]\[
b = \sin(23^\circ) \times 30 \, cm \approx 11.8 \, cm \quad (\text{as found in the answer})
\][/tex]
4. Calculate the area of the triangle:
Using the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Here, the base is the adjacent leg (27.6 cm) and the height is the opposite leg (11.8 cm):
[tex]\[
\text{Area} = \frac{1}{2} \times 27.6 \, cm \times 11.8 \, cm \approx 162.3 \, cm^2
\][/tex]
We round the result to the nearest tenth.
5. Conclusion:
The approximate area of the triangle is [tex]\(162.3 \, cm^2\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{161.8 \, cm^2} \][/tex]
