The Cosine Rule and Triangle Area Formula
Learn the cosine rule and ½ab sinC area formula for IB MYP Extended Maths Year 5. Know when to use sine rule vs cosine rule in non-right triangles.
When to Use the Cosine Rule
The cosine rule is used when you know:
- All three sides of a triangle (SSS) and need to find an angle, or
- Two sides and the included angle (SAS) and need to find the third side
These are the situations where the sine rule cannot be applied directly.
The Cosine Rule
To find a missing side: a² = b² + c² − 2bc cos A
To find a missing angle, rearrange for cos A: cos A = (b² + c² − a²) / 2bc, then apply cos⁻¹.
Area of a Triangle Using ½ab sinC
When you know two sides and the included angle, the area of any triangle is:
Area = ½ ab sin C
This formula replaces the need for a perpendicular height. It is particularly useful in non-right triangles where the height is not given or is difficult to calculate.
Choosing Between Sine Rule and Cosine Rule
- Sine rule: Use with AAS, ASA, or SSA
- Cosine rule: Use with SAS or SSS
- Area formula: Use with SAS
In multi-step problems, you may need to apply one rule to find a missing element and then switch to the other. The ability to choose correctly is assessed in Criterion A and Criterion B (investigating patterns and reasoning).
Common Mistakes
- Using the cosine rule with the wrong angle — angle C must be the included angle between sides a and b
- Sign errors when rearranging for cos A, especially with the negative term −2bc cos A
- Using ½ base × height when the height is not known — use ½ab sinC instead
- Applying the sine rule in an SAS situation where it does not directly apply
MYP Question Style
These questions frequently appear in Criterion D contexts: land surveying, navigation, or architecture problems. You will be expected to select and justify your method, not just execute a calculation. Showing clear reasoning at each decision point earns marks in higher achievement bands.