Sine Rule

Sine Rule: how to use it

The sine rule links the sides and angles of a triangle when you know an angle–side pair and need to find another side or angle. It is especially useful in non-right triangles.

1) Match the opposite pairs

Identify each side and the angle directly opposite it. The rule compares these pairs:

• side sin(opposite angle)

2) Set up the proportion

Write the known pair and the unknown pair in the same fraction form. A common setup is:

• (\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}) Use only the values you know and the one you want to find.

3) Rearrange carefully

If a side is missing, cross-multiply and isolate it. If an angle is missing, first isolate the sine value, then use the inverse sine if appropriate.

4) Simplify the final answer

Leave exact values in simplest form when possible. If the answer involves a fraction or surd, reduce it fully.

5) Check your result

• The angle sum in a triangle should be 180b0.
• A side should be reasonable compared with the other given sides.
• If you found an angle, make sure it matches the triangle’s shape and the opposite side size.

A good habit is to label the triangle first before writing the sine rule. That prevents mixing up opposite pairs.