Given Side Lengths:
a = ?
b = 17.23
c = 10.86
Given Angles (in degrees):
A = ?
B = 101
C = ?
Note: We have highlighted the angle(s) and side that we need to find. Let's draw a rough sketch of the triangle:
The sin rule is:
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]From the information given, we can write a ratio using the sine rule:
[tex]\frac{\sin101}{17.23}=\frac{\sin C}{10.86}[/tex]We can cross multiply and solve for the value of angle C:
[tex]\begin{gathered} \frac{\sin101}{17.23}=\frac{\sin C}{10.86} \\ 10.86\sin 101=17.23\sin C \\ \sin C=\frac{10.86\sin101}{17.23} \\ \sin C=0.6187 \\ C=\sin ^{-1}(0.6187) \\ C=38.22\degree \end{gathered}[/tex]Now, we know 3 angles in a triangle add up to 180 degrees. Thus, we can solve for the Angle A:
[tex]\begin{gathered} A+B+C=180 \\ A+101+38.22=180 \\ A+139.22=180 \\ A=180-139.22 \\ A=40.78\degree \end{gathered}[/tex]We can write another ratio using the sin rule and find the side length, a. Shown below:
[tex]\frac{\sin101}{17.23}=\frac{\sin40.78}{a}[/tex]Let's do a little algebra and solve for the side length, a:
[tex]\begin{gathered} \frac{\sin101}{17.23}=\frac{\sin40.78}{a} \\ a\sin 101=17.23\sin 40.78 \\ a=\frac{17.23\sin 40.78}{\sin 101} \\ a=11.46 \end{gathered}[/tex]Answer[tex]\begin{gathered} a=11.46\text{ cm} \\ A=40.78\degree \\ C=38.22\degree \end{gathered}[/tex]
