Sine Rule

1Summary 重點摘要

Learn another universal formula for any triangle, especially when you know angles and opposite sides.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What the sine rule doesThe sine rule, like the cosine rule, works for any triangle — right-angled, acute or obtuse. It shines when a question gives an angle with its opposite side, plus one more angle or side. Sketch the triangle first and label each angle with its opposite side before calculating.
2. Step 2: The formula and the pairingThe formula is a/sin A = b/sin B = c/sin C, where a is the side opposite angle A, b opposite B and c opposite C. In words: each side divided by the sine of its opposite angle gives the same value. Always pair each side with its own angle — a with A, b with B, c with C — because one wrong pairing ruins the whole equation.
3. Step 3: How to find a sideKnowing one angle with its opposite side, plus a second angle, is enough to find another side. For example, given a=10, angle A=30° and angle B=45°, write the proportion 10/sin30° = b/sin45° with the known pair on the left and the unknown b on the right. In such 'two angles and one side' questions the third angle is already fixed (180° − 30° − 45° = 105°), and the sine rule is the most direct method.
4. Step 4: Worked calculationContinuing the example: 10/sin30° = b/sin45°, with sin30° = 0.5 and sin45° = √22 ≈ 0.707. Substituting gives 100.5 = b/(√22), so 20 = 2b/√2 and b = 20 × √22 = 10√2 ≈ 14.1. As a sanity check, b is longer than a because angle B (45°) is larger than angle A (30°); sides grow with their opposite angles, so the answer makes sense.
5. Step 5: How to find an angleGiven two sides and the angle opposite one of them, the rule can be rearranged to find another angle. For example, with a=10, b≈14.1 and A=30°, write sin B = b × sin A / a = 14.1 × 0.5 ÷ 10 = 0.705, then use the inverse sine key to get B ≈ 44.8°. Be alert: such questions sometimes allow a second (obtuse) solution, so judge from the context and the angle sum which answer fits.
6. Step 6: Choosing sine or cosine ruleChoose by the given data: two angles plus any side, or two sides plus the angle opposite one of them, call for the sine rule. Two sides with the included angle, or three sides, call for the cosine rule. In short, the sine rule needs one complete angle-side pair to start, while the cosine rule does not. Picking the wrong rule wastes time and leads to an equation you cannot finish, so classify the given conditions first.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Works for any triangle
• a/sin A = b/sin B = c/sin C
• Use when know angles and opposite sides
• Match side to opposite angle, don't mismatch
• Complements cosine rule