Sine, Cosine Rule & Area of Triangles
Master non-right-angled triangle trigonometry for IGCSE Extended Maths
What You'll Learn
- The Sine Rule: Find missing sides or angles when you have opposite pairs (a side and the angle across from it)
- The Cosine Rule: Find missing sides or angles when you know two sides and the angle between them, or all three sides
- Area Formula: Calculate the area of any triangle when you know two sides and the included angle
- Decision-Making: Choose the right tool for each question and combine rules when needed
- The Ambiguous Case: Why the sine rule can sometimes give two answers (and how to fix it)
The Sine Rule
What is the Sine Rule?
The sine rule is your tool for finding missing sides or angles in a non-right-angled triangle, but only when you have an opposite pair — that is, you know a side and the angle directly across from it in the triangle.
Think of it like this: in any triangle, the ratio of a side to the sine of its opposite angle is constant. This is incredibly powerful because it means if you know one complete pair, you can find any other side or angle.
Key insight: Notice that angles use CAPITAL letters and sides use lowercase letters. The side a is opposite angle A, side b is opposite angle B, and so on. This is not just notation — it's a crucial rule that saves you from making mistakes. Always label your triangle this way first.
When Do I Use the Sine Rule?
Use the sine rule when you have:
- Two sides and an angle opposite one of them, and you want to find the angle opposite the other side, OR
- Two angles and a side opposite one of them, and you want to find the side opposite the other angle
Finding Missing Lengths with the Sine Rule
If you're looking for a side length, use the sine rule with sides in the numerators:
The steps are simple:
- Label your triangle with capital letters for angles and lowercase for opposite sides
- Identify which side you know, which angle you know, and which side you want to find
- Set up the equation using just two parts of the formula
- Rearrange and solve
✎ Worked Example: Finding a Side
In triangle ABC, side BC = 12.3 cm, angle BCA = 27°, side AB = 8.1 cm. Find the length of side AC.
Step 1: Relabel for clarity. We have:
- Side opposite to A (which is BC) = a = 12.3 cm
- Angle A = angle BCA = 27°
- Side opposite to B (which is AC) = b = ? (what we want)
- Side opposite to C (which is AB) = c = 8.1 cm
Step 2: We have side a and angle A, plus side c. We want side b. But wait — we need the angle opposite to b, which is angle B.
Using sine rule: a / sin A = c / sin C
First find angle B (or use a different pair if angle B is known). Actually, let's use: b / sin B = a / sin A
But we need angle B first. Let me reconsider: we have a pair (a = 12.3, A = 27°) and another side c = 8.1. We want side b = AC.
Step 3: Actually, looking at the original triangle description again: AB = 8.1, BC = 12.3, angle BCA = 27°. So we need to find angle ABC (which is the angle at B).
Step 4: