Not all triangles have a right angle, but our definitions for sine, cosine, and tangent are based on a right angle. As shown in figure 1 we can construct a right angle for any triangle and thereby use our definitions. The line segment, “opp” created a right angle. $$\sin A=\frac{opp}{c}$$ and $$\sin C=\frac{opp}{a}.$$ Therefore, $$\frac{\sin A}{a}=\frac{\sin C}{c}.$$ Similarly, we can show that $$\frac{\sin A}{a}=\frac{\sin B}{b}.$$ To do so, just drop a perpendicular from vertex $C$.
Figure 1: Triangle $ABC$ without a right angle. The line segment, “opp” created a right angle. $\sin A=\frac{opp}{c}$ and $\sin C=\frac{opp}{a}$. Therefore, $\frac{\sin A}{a}=\frac{\sin C}{c}$. Similarly, we can show that $\frac{\sin A}{a}=\frac{\sin B}{b}$.
