To find the area of the triangle when given side [tex]\( a = 19.2 \)[/tex], angle [tex]\( A = 53.8^\circ \)[/tex], and angle [tex]\( C = 65.4^\circ \)[/tex], we can follow these steps:
1. Convert the angles from degrees to radians:
- [tex]\( A = 53.8^\circ \)[/tex]
[tex]\[
A_\text{rad} \approx 0.939 \, \text{radians}
\][/tex]
- [tex]\( C = 65.4^\circ \)[/tex]
[tex]\[
C_\text{rad} \approx 1.141 \, \text{radians}
\][/tex]
2. Find the missing angle [tex]\( B \)[/tex]:
Since the sum of the angles in a triangle is [tex]\( 180^\circ \)[/tex]:
[tex]\[
B = 180^\circ - A - C = 180^\circ - 53.8^\circ - 65.4^\circ = 60.8^\circ
\][/tex]
Convert [tex]\( B \)[/tex] to radians:
[tex]\[
B_\text{rad} \approx 1.061 \, \text{radians}
\][/tex]
3. Use the Law of Sines to find side [tex]\( b \)[/tex]:
[tex]\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)}
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = a \cdot \frac{\sin(B_\text{rad})}{\sin(A_\text{rad})} \approx 20.77
\][/tex]
4. Use the Law of Sines to find side [tex]\( c \)[/tex]:
[tex]\[
\frac{a}{\sin(A)} = \frac{c}{\sin(C)}
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = a \cdot \frac{\sin(C_\text{rad})}{\sin(A_\text{rad})} \approx 21.63
\][/tex]
5. Calculate the area of the triangle using the formula:
[tex]\[
\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C_\text{rad})
\][/tex]
[tex]\[
\text{Area} \approx 181.29
\][/tex]
6. Round the area to the nearest tenth:
[tex]\[
\text{Area} \approx 181.3
\][/tex]
Therefore, the area of the triangle, rounded to the nearest tenth, is [tex]\( 181.3 \)[/tex].
