The Sine Rule: Solving Non-Right-Angled Triangles

When to Use the Sine Rule

The sine rule applies to any triangle — not just right-angled ones. You use it when you know either:

• Two angles and one side (AAS or ASA), or
• Two sides and a non-included angle (SSA)

If the triangle contains a right angle, use basic trig ratios instead — they are simpler.

The Sine Rule

For a triangle with sides a, b, c opposite to angles A, B, C respectively:

a / sin A = b / sin B = c / sin C

To find a missing side, set up two of these fractions and cross-multiply. To find a missing angle, rearrange to isolate sin of the unknown angle, then apply sin⁻¹.

The Ambiguous Case (SSA)

When you are given two sides and a non-included angle, there may be two possible triangles — this is the ambiguous case of the sine rule. At MYP Extended level, questions will usually indicate which solution is valid through context (e.g. an angle must be obtuse or acute). Be aware that this situation exists and check whether a second solution is geometrically possible.

Step-by-Step for Finding a Side

1. Label the triangle with sides and opposite angles using consistent notation.
2. Write the sine rule with the unknown side in the numerator.
3. Substitute the known angle-side pair into the denominator.
4. Multiply both sides to isolate the unknown.

Common Mistakes

• Using the sine rule on a right-angled triangle when basic ratios would be faster and less error-prone
• Pairing a side with the wrong angle (side a must be opposite angle A)
• Ignoring the ambiguous case when SSA conditions are present