SOLUTION
To solve, this problem we will use the sine rule:
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
First, let us solve for B, We will relate:
[tex]\begin{gathered} \frac{b}{\sin B}=\frac{c}{\sin C} \\ since\text{ the c part of the formula is complete} \end{gathered}[/tex][tex]\begin{gathered} \frac{5}{\sin B}=\frac{11}{\sin 120} \\ \text{Cross multiply} \\ 5\times\sin 120=11\times\sin B \\ \end{gathered}[/tex][tex]\begin{gathered} 4.33013=11\sin B \\ \frac{4.33013}{11}=\sin B \\ 0.393648=\sin B \\ \sin ^{-1}(0.393648)=B \\ 23.1817^o=B \\ 23.2^o(to\text{ the nearest tenth)=B} \end{gathered}[/tex]
B = 23.2 degrees.
To find A, we will use the sum of angles in a triangle:
[tex]A+B+C=180^o[/tex][tex]\begin{gathered} A+23.2+120=180 \\ A=180-120-23.2 \\ A=36.8^o \end{gathered}[/tex]
A = 36.8 degrees.
To find a, we will use the sin rule again.
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \frac{a}{\sin36.8}=\frac{11}{\sin 120} \end{gathered}[/tex]
Cross multiply:
[tex]\begin{gathered} a\times\sin 120=11\times\sin 36.8 \\ a=\frac{11\times\sin 36.8}{\sin 120} \\ a=\frac{11\times0.599024}{0.866025} \\ a=\frac{6.589264}{0.866025} \end{gathered}[/tex][tex]\begin{gathered} a=7.6086 \\ a=7.6(to\text{ the nearest tenth)} \end{gathered}[/tex]
a=7.6
Final answers:
A=36.8 degrees, B=23.2 degrees, a=7.6
