The Law of Sines
It's an equation that relates the lengths of the sides of a triangle with the sines of its angles.
For the given triangle, the equation is:
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
a.
We are given: m/_A = 80°, m/_C = 33°, and c=45.1.
We use the first and the last part of the equation above:
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \frac{a}{\sin80^o}=\frac{45.1}{\sin 33^o} \end{gathered}[/tex]
Solving for a:
[tex]a=\sin 80^o\cdot\frac{45.1}{\sin33^o}[/tex]
Calculating:
[tex]a=0.9848\cdot\frac{45.1}{0.5446}=81.6[/tex]
a = 81.6
b.
Now we are given m/_B=48°, m/_C=26°, a=19.9
Since we are required to calculate the value of c and we are not given the value of the angle A, we first determine it recalling the sum of angles of a triangle is 180°, thus:
m/_A= 180° - 48° - 26° = 106°
Now we apply the equation:
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \frac{19.9}{\sin 106^o}=\frac{c}{\sin 26^o} \end{gathered}[/tex]
Solving for c:
[tex]\begin{gathered} c=\sin 26^o\cdot\frac{19.9}{\sin106^o} \\ c=0.4384\cdot\frac{19.9}{0.9613}=9.08 \end{gathered}[/tex]
c = 9.08
