Given two angles and one side of the triangle, you have to find the missing angle and the missing sides.
To do it, you can follow the steps.
Step 1: Find the missing angle.
Knowing that the sum of interior angles of a triangle is 180°, you can find A.
A + B + C = 180°
Knowing that B = 42° and C = 100°, you can substitute them in the equation and find A.
A + 42 + 100 = 180
A + 142 = 180
Adding - 142 to both sides:
A + 142 - 142 = 180 - 142
A + 0 = 38
A = 38°
Step 2: Find the missing sides.
Since all the angles and only one side are known, you can use the sen rule.
[tex]\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}[/tex]Substituting the values, you have:
[tex]\frac{a}{\sin(38)}=\frac{165}{\sin(42)}=\frac{c}{\sin(100)}[/tex]First, let's compare sides a and b:
[tex]\frac{a}{\sin(38)}=\frac{165}{\sin(42)}[/tex]Multiplying both sides bi sin(38):
[tex]\begin{gathered} \frac{a}{\sin(38)}\cdot\sin (38)=\frac{165}{\sin(42)}\cdot\sin (38) \\ a=165\cdot\frac{\sin(38)}{\sin(42)} \end{gathered}[/tex]And solving the equation:
[tex]\begin{gathered} a=165\cdot\frac{0.6157}{0.6691} \\ a=151.8 \end{gathered}[/tex]Now, let's find c by comparing b and c:
[tex]\frac{c}{\sin(100)}=\frac{165}{\sin(42)}[/tex]Multiplying both sides by sin(100) and solving the equation.
[tex]\begin{gathered} \frac{c}{\sin(100)}\cdot\sin (100)=\frac{165}{\sin(42)}\cdot\sin (100) \\ c=165\cdot\frac{\sin (100)}{\sin (42)} \\ c=165\cdot\frac{0.9848}{0.6691} \\ c=242.8 \end{gathered}[/tex]Done! You found the missing sides and angles.
As you can see, there is only one possible solution.
And the solution is:
a = 151.8; A = 38°
b = 165; B = 42°
c = 242.8; C = 100°