Math Trigonometry Law of sines

Law of sines

The sine rule is applicable in any triangle, if either

  • one side and two angles
  • or
  • two sides and one angle are given.

In the second case, the angle must be opposite to one of the two given sides. Otherwise, you need the cosine rule.



In any triangle, two lengths are related to each other like the opposite sine values.

In our triangle ABC, the sine rule says:

  • $\frac{a}{b}=\frac{\sin(\alpha)}{\sin(\beta)}$

  • $\frac{b}{c}=\frac{\sin(\beta)}{\sin(\gamma)}$

  • $\frac{c}{a}=\frac{\sin(\gamma)}{\sin(\alpha)}$

  • $\frac{a}{\sin(\alpha)}=\frac{b}{\sin(\beta)}=\frac{c}{\sin(\gamma)}$


Given is a triangle with $a=7$, $c=4$ and $\gamma=30^\circ$. Calculate the angle $\alpha$.

  1. Find the right formula

  2. Change the formula

  3. Insert and calculate