Explanation:
Taking into account the law of cosines, we can write the following equation:
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
Then, replacing the values for a, b, and c, we get:
[tex]\begin{gathered} 5^2=2^2+6^2-2(2)(6)\cos A \\ 25=4+36-24\cos A \\ 25=40-24\cos A \end{gathered}[/tex]
Solving for cos A, we get:
[tex]\begin{gathered} 25-40=40-24\cos A-40 \\ -15=-24\cos A \\ \frac{-15}{-24}=\frac{-24\cos A}{-24} \\ 0.625=\cos A \end{gathered}[/tex]
Therefore, the value of angle A is:
[tex]\begin{gathered} \cos ^{-1}(0.625)=A \\ 51.3=A \end{gathered}[/tex]
Now, we can use the law of sines, we can write the following equation:
[tex]\frac{\sin B}{b}=\frac{\sin A}{a}[/tex]
So, replacing the values and solving for B, we get:
[tex]\begin{gathered} \frac{\sin B}{2}=\frac{\sin 51.3}{5} \\ \sin B=\frac{\sin 51.3}{5}\times2 \\ \sin B=0.3121 \\ B=\sin ^{-1}(0.3121) \\ B=18.2 \end{gathered}[/tex]
In the same way, Angle C is equal to:
[tex]\begin{gathered} \frac{\sin C}{c}=\frac{\sin A}{a} \\ \frac{\sin C}{6}=\frac{\sin 51.3}{5} \\ \sin C=\frac{\sin51.3}{5}\times6 \\ \sin C=0.9365 \\ C=\sin ^{-1}(0.9365) \\ C=69.5 \end{gathered}[/tex]
So, the answers are:
A = 51.3°
B = 18.2°
C = 69.5°
