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.

!

Remember

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

In our triangle ABC, the sine rule says:

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

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

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

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

Example

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

1. Find the right formula

$\frac{a}{c}=\frac{\sin(\alpha)}{\sin(\gamma)}$
2. Change the formula

$\frac{a}{c}=\frac{\sin(\alpha)}{\sin(\gamma)}\quad|\cdot\sin(\gamma)$
$\sin(\alpha)=\frac{a}{c}\cdot\sin(\gamma)$
3. Insert and calculate

$\sin(\alpha)=\frac{7}{4}\cdot\sin(30^\circ)$
$\alpha=\sin^{-1}(\frac{7}{4}\cdot\sin(30^\circ))\approx61^\circ$