This question was uploaded on 28/11/21 on social media accounts.
Solution:
By given conditions, all four terms are part of increasing A.P:
Let the first term be 'a' and the common difference is d:
\[a_1 = a\]
\[a_2 = a + d\]
\[a_3 = a + 2d\]
\[a_4 = a + 3d\]
Given that first term is smaller than last term:
\[\Rightarrow a_1 < a_2 < a_3 < a_4\]
Here 60 degrees angle is not at the opposite of the smallest or the largest side:
\[\Rightarrow A = 60^\circ\]
By using the cosine rule:
\[\cos60 = \frac{a^2+(a+2d)^2-(a+3d^2)}{2(a)(a+3d)}\]
\[\Rightarrow a=5d\]
By substituting these values:
\[a_1:a_3:a_4 = 5:7:8\]
Now by using the sine rule:
\[\frac7{\sin 60} = \frac8{\sin B} = \frac5{\sin C}\]
\[\color{red} { \Rightarrow \sin A = \frac{\sqrt3}{2}}\]
\[\color{red} { \Rightarrow \sin B = \frac{8\sqrt3}{14}}\]
\[\color{red} { \Rightarrow \sin C = \frac{5\sqrt3}{14}}\]
