Key takeaways
- Use the Sine Rule when you know a matching pair of an angle and its opposite side (AAS, ASA, or SSA).
- Use the Cosine Rule when you have all three sides (SSS) or two sides with the angle trapped between them (SAS).
- The Sine Rule is algebraically simpler, while the Cosine Rule unlocks triangles where no opposite angle-side pair is known.
- Beware of the ambiguous case (SSA) with the Sine Rule, which can yield two possible triangles.
- Always check that your triangle angles add up to exactly 180° before and after calculation.
In this guide
What is the difference between the Sine Rule and Cosine Rule?
The Sine Rule states that the ratio of each side length to the sine of its opposite angle is constant (a/sin(A) = b/sin(B) = c/sin(C)). The Cosine Rule generalizes the Pythagorean theorem to non-right triangles (c² = a² + b² − 2ab cos(C)). You choose between them based on what you are given: use the Sine Rule when you have a matching angle-side pair, and the Cosine Rule when you have three sides (SSS) or two sides and the angle between them (SAS).
Decision Guide: Which rule should you use?
Look at the three measurements you are given and match them to this table:
| Given Measurements | Triangle Type | Rule to Use | What You Find First |
|---|---|---|---|
| Two angles and one side | AAS or ASA | Sine Rule | Third angle (180° − A − B), then remaining sides |
| Two sides and an opposite angle | SSA | Sine Rule | Second angle (check for ambiguous case) |
| Two sides and the included angle | SAS | Cosine Rule | Third side opposite the known angle |
| All three sides | SSS | Cosine Rule | Any angle (start with the largest side) |
The Sine Rule Explained
The Sine Rule relates sides and opposite angles across the whole triangle:
When calculating an unknown side, keep side lengths on top as written above. When calculating an unknown angle, flip the fractions upside down so the angle sits in the numerator:
Worked Example: Finding a side with the Sine Rule
Suppose a triangle has angle A = 40°, angle B = 75°, and side a = 8 cm. Find side b.
- 1Identify the pair: We know angle
A = 40°and sidea = 8. This gives our complete ratio:8 / sin(40°) ≈ 12.446. - 2Set up the equation:
b / sin(75°) = 8 / sin(40°). - 3Multiply across:
b = 8 × sin(75°) / sin(40°). - 4Calculate:
b = 8 × 0.9659 / 0.6428 ≈ 12.02 cm.
To verify your working on similar problems, use the interactive sine rule calculator.
The Cosine Rule Explained
The Cosine Rule is needed when you do not have any matching angle-side pair. The standard formula finds an unknown side c opposite angle C:
If you need to find an angle when all three sides a, b, and c are known, rearrange the formula to solve for cos(C):
Worked Example: Finding a side with the Cosine Rule (SAS)
A triangle has sides a = 7 m, b = 10 m, and included angle C = 60°. Find side c.
- 1Substitute into the formula:
c² = 7² + 10² − 2(7)(10) cos(60°). - 2Evaluate powers and products:
c² = 49 + 100 − 140(0.5000). - 3Subtract:
c² = 149 − 70 = 79. - 4Square root:
c = √79 ≈ 8.89 m.
To solve triangles with different side configurations, explore the cosine rule calculator.
Common Traps to Avoid
- Calculator Angle Mode: Ensure your calculator is set to Degrees (DEG), not Radians (RAD), when entering degree angles.
- Order of Operations in the Cosine Rule: In
a² + b² − 2ab cos(C), you must multiply2 × a × b × cos(C)before subtracting froma² + b². A very common mistake is calculating(a² + b² − 2ab) × cos(C), which produces an incorrect result.
Summary Checklist
Before starting any non-right triangle question: (1) Check if you have an angle and its opposite side — if yes, pick the Sine Rule. (2) If you have two sides trapping an angle (SAS) or all three sides (SSS), pick the Cosine Rule. (3) Verify your solution using our free step-by-step trigonometry solvers.
Frequently asked questions
When should I use the Sine Rule instead of the Cosine Rule?
Use the Sine Rule whenever you know an angle and its opposite side (AAS, ASA, or SSA). Use the Cosine Rule when you do not have any matching opposite pair (SAS or SSS).
Can the Sine Rule be used on right-angled triangles?
Yes, because sin(90°) = 1, the Sine Rule simplifies directly to standard trigonometry (SOH-CAH-TOA). However, basic SOH-CAH-TOA is faster for right triangles.
What is the ambiguous case of the Sine Rule?
When given two sides and a non-included acute angle (SSA), there may be two distinct triangles that satisfy the conditions because sin(θ) = sin(180° − θ). Always test whether the supplementary angle fits.
How do I rearrange the Cosine Rule to find an angle?
cos(C) = (a² + b² − c²) / (2ab). Take the inverse cosine (arccos) of the result to find the angle C.
