Sine and Cosine Rules
This topic covers essential rules for solving any triangle, not just right-angled ones. These formulae allow you to find unknown sides and angles when given sufficient information and are applicable in both 2D and 3D problems.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The Sine Rule, Cosine Rule, and Area formula apply to ALL triangles.
- Standard notation is crucial: Angle A is opposite side a, Angle B is opposite side b, etc.
- The 'ambiguous case' of the Sine Rule can arise when you are given two sides and a non-included angle (ASS), potentially leading to two valid triangles.
- 3D problems almost always simplify to solving a 2D triangle within the 3D shape. Your first step should be to identify and sketch this triangle.
- The Cosine Rule is a generalisation of Pythagoras' Theorem, with an extra term to account for non-right angles.
Diagram
Formulae
Area = 1/2 a b sin(C) To find the area of a triangle when you know two sides and the angle between them (the included angle).
a / sin(A) = b / sin(B) = c / sin(C) When you know a side and its opposite angle, plus one other side or angle. Use it to find a missing side or a missing angle.
a2 = b2 + c2 - 2 b c cos(A) To find the third side when you know two sides and the included angle. Or, rearranged, to find any angle when you know all three sides.
cos(A) = (b2 + c2 - a2) / (2 b c) The rearranged Cosine Rule, used to find an angle when all three side lengths are known.
Definitions
- Included Angle
- The angle formed between two specified sides of a triangle. For example, angle C is the included angle between sides a and b.
- Ambiguous Case (Angle-Side-Side)
- A situation when using the Sine Rule to find an angle where two different valid triangles can be constructed from the given information. This occurs because sin(x) = sin(180° - x), so there are two possible angles less than 180° for any given sine value between 0 and 1.
Worked example
In triangle PQR, the side PQ has length 8 cm, and the side PR has length 10 cm. The angle QPR is 120°. What is the length of the side QR?
- 1
Identify the knowns:
We have two sides (p = 10, q = 8) and the included angle (R = 120°).
We need to find the third side, r (which is QR).
- 2
Choose the correct formula.
Since we have two sides and the included angle, we must use the Cosine Rule:
r2 = p2 + q2 - 2pq cos(R) - 3
Substitute the values into the formula:
r2 = 102 + 82 - 2(10)(8)cos(120°) - 4
Recall the exact value for cos(120°).
This is in the second quadrant, so it's negative.
cos(120°) = -cos(60°) = -1/2 - 5
Calculate r2:
r2 = 100 + 64 - 160(-1/2) = 164 + 80 = 244 - 6
Solve for r:
r = √(244)We can simplify this:
√(244) = √(4 × 61) = 2 × √(61) cm
Answer: 2 × √(61) cm
Common mistakes
- ×Using the incorrect formula for the given information, e.g., trying to use the Sine Rule with two sides and an included angle.
- ×Forgetting that cos(A) is negative for obtuse angles (between 90° and 180°), which changes the sign of the '-2bc cos(A)' term in the Cosine Rule.
- ×When finding an angle with the Sine Rule, only giving the acute solution (from the calculator) and failing to consider the obtuse possibility (180° - angle), which is the essence of the ambiguous case.
- ×Making arithmetic errors when squaring numbers or applying the order of operations in the Cosine Rule.
No-calculator tips
- ✓Memorise the exact trigonometric values for 30°, 45°, 60°, and their equivalents in other quadrants (e.g., 120°, 135°, 150°). Questions will always use these to allow for non-calculator solutions.
- ✓When using the Cosine Rule, deal with the `2bc cos(A)` term separately before adding or subtracting it from `b2 + c2` to avoid errors.
- ✓If a question asks for sin2(x) or a side length squared, it's a hint that you might not need to compute a difficult square root at the end.
- ✓Always draw a clear, reasonably-scaled diagram. This can help you spot if an angle should be obtuse or acute, which is vital for the ambiguous case.