Sine rule

Martin McBride

The sine rule is a trigonometry formula that relates the sides and angles of a triangle. It can be used to solve a triangle if we know either:

  • Two angles and any side of the triangle.
  • Two sides and any angle except the angle enclosed by the two sides.

For other cases you will need to use the cosine rule.

The rule applies to any triangle, not just right angled triangles.

This rule is also covered in this video on youtube:

Labelling the triangle

It is important to label the triangle correctly, otherwise the rule won't work! We name the angles A, B and C, and we name the sides a, b and c:

The important thing to remember is that each angle is opposite the side of the same name:

  • Angle A is opposite side a.
  • Angle B is opposite side b.
  • Angle C is opposite side c.

The sine rule

The sine rule tells us that:

$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$

This is a short way of writing these three equations:

$$ \frac{a}{\sin A} = \frac{b}{\sin B} $$

$$ \frac{a}{\sin A} = \frac{c}{\sin C} $$

$$ \frac{b}{\sin B} = \frac{c}{\sin C} $$

We can also flip this equation:

$$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$

This second form can be useful if you want to find an angle.