When to use the cosine rule: choose between the sine rule and the cosine rule from the given information
The sine rule and the cosine rule have clear roles: if the question gives two angles (or one angle and its opposite side), use the sine rule; if it gives two sides and the included angle, or all three sides, use the cosine rule. The key is which three pieces of information the question gives, not which formula is easier to memorise. Below is a three-step method, then worked examples, then common mistakes and revision advice. The method here is general study advice, not official exam guidance.
This article was put together automatically from general study experience and is for reference only. For exam and school arrangements, check the latest information from the organisation concerned.
- List the known sides and angles first, then decide which rule to use.
- With two angles, or one angle and its opposite side, use the sine rule.
- With two sides and the included angle, or three sides, use the cosine rule.
- A negative cosine means an obtuse angle, not a mistake.
- The largest angle is opposite the longest side; use this to check answers.
First ask: which three pieces of information does the question give?
Triangle problems usually involve three sides a, b, c and three angles A, B, C, where a is opposite A, b is opposite B and c is opposite C. Any rule can only be used once you know three pieces of information, so the first step is not to recall a formula but to write down what the question gives, marking each item as a side or an angle.
Once you have written them down, check two conditions. If the given information includes two angles, or one angle and its opposite side, the sine rule can be used directly. If it includes two sides and the included angle, or all three sides, the cosine rule is the one to use. If the three pieces fit neither, you usually need the angle sum of a triangle (A + B + C = 180°) or another condition to fill the gap before you can calculate.
The benefit of this habit is that choosing the formula becomes something you can check. In an exam time is tight, and choosing a rule by feel is the easiest way to go wrong. Listing the given information first cuts down on second-guessing.
The sine rule: for cases with an angle and its opposite side
The sine rule is a / sin A = b / sin B = c / sin C. Its structure pairs each side with its opposite angle, so as long as you have one complete side-and-opposite-angle pair, plus another angle or side, you can find the unknown parts.
There are two typical cases. First, two angles and one side are given: for example A = 40°, B = 65°, a = 8. Use the angle sum to get C = 75°, then use the sine rule to find b and c. Second, two sides and the angle opposite one of them are given: for example a = 7, b = 9, A = 35°. Use the sine rule to find sin B, then decide the possible values of B.
The second case needs care, because sine is not one-to-one between 0° and 180°: the same sine value can belong to an acute angle and an obtuse angle. This is the ambiguous case. Check whether the two angles still add up to less than 180°, and keep the answers the question asks for.
Conversely, if the question gives two sides and the included angle, there is no complete side-and-opposite-angle pair among those three pieces, so the sine rule cannot get started. Use the cosine rule instead.
The cosine rule: for an included angle or three sides
The cosine rule is c² = a² + b² − 2ab cos C, and the other two forms just rotate the letters. You can think of it as an extension of Pythagoras' theorem: when C = 90°, cos C = 0 and the formula becomes c² = a² + b².
The first case is two sides and the included angle. For example a = 6, b = 10, C = 60°. Substitute directly to find c: c² = 36 + 100 − 2 × 6 × 10 × cos 60° = 136 − 60 = 76, so c = √76, about 8.72. Once you have the third side, if you want another angle you can use the cosine rule again, or use the sine rule on the smaller angle to avoid the ambiguous case.
The second case is three sides, finding an angle. For example a = 5, b = 7, c = 9, and you want C. Rearrange to cos C = (a² + b² − c²) / (2ab) = (25 + 49 − 81) / (2 × 5 × 7) = −7 / 70 = −0.1, so C is about 95.7°. A negative cosine means an obtuse angle, and the sign itself is a useful check. Likewise, to find angle A use cos A = (b² + c² − a²) / (2bc); to find angle B use cos B = (a² + c² − b²) / (2ac).
The cosine rule does not need a side paired with its opposite angle, so when the information is two sides and the included angle, or three sides, it is almost always the direct choice.
The three-step method with worked examples
Turn the checks above into three steps. Step 1: list the known sides and angles and confirm each side is paired with its opposite angle. Step 2: check for two angles, or one angle and its opposite side. If present, use the sine rule. Step 3: check for two sides and the included angle, or three sides. If present, use the cosine rule. If neither applies, use the angle sum or another condition to fill in the three pieces, then go back to step 1.
Example 1: A = 50°, B = 70°, a = 9, find b. This is two angles and one side, so use the sine rule. First C = 60°, then b = 9 sin 70° / sin 50°, about 11.04.
Example 2: a = 8, b = 5, C = 120°, find c. This is two sides and the included angle, so use the cosine rule: c² = 64 + 25 − 2 × 8 × 5 × cos 120° = 89 + 40 = 129, so c is about 11.36. Note that cos 120° is negative, so the minus sign turns into a plus. This is a common trap.
Example 3: the three sides are 4, 6 and 7, find the largest angle. The largest angle is opposite the longest side, so let C be opposite 7: cos C = (16 + 36 − 49) / (2 × 4 × 6) = 3 / 48 = 0.0625, so C is about 86.4°. Use the cosine rule here because all three sides are known but no angle is given.
Common mistakes and checking habits
The first common mistake is seeing two sides and assuming the sine rule applies, ignoring the need for a side-and-opposite-angle pair. The second is forgetting to handle a negative cosine in the cosine rule, or writing the sign of − 2ab cos C wrongly. The third is using the sine rule to find an angle without checking the ambiguous case, which leads to a missing answer or an angle that does not fit the question.
Checking is simple: the largest angle should be opposite the longest side; the angles must add up to 180°; the cosine value must lie between −1 and 1. If your cos value falls outside that range, an earlier step has gone wrong. Making these checks a habit saves more time than redoing the whole question.
As for whether you need tutoring, first work out where you are stuck. If you simply cannot remember the formulas, free short video lessons to revisit the topic are enough. If your thinking about the given information is muddled, you need practice on the step of choosing the rule, not blind drilling of lots of questions.
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Revision plan for HKDSE students
Trigonometry questions often put the sine rule, the cosine rule and the area formula in one question, so it helps to build a single table matching given information to formula, with an example you have worked out beside each formula. The aim is to make choosing the rule your first reaction on seeing a question, not something you work out on the spot.
When practising, sort the questions on purpose: do a batch using only the sine rule, then a batch using only the cosine rule, then mixed questions. Sorted practice makes the skill of reading the conditions steady; mixed practice checks whether you can really tell them apart. Each time you get one wrong, note whether you chose the wrong rule or made a calculation error. The remedy is different for each.
Building a solid base in angles, the angle sum of a triangle and Pythagoras' theorem in primary and junior secondary makes learning these two rules in senior secondary much smoother. To revisit topics by year, see the year pages on this site.
The HKDSE maths papers require several types of question within a time limit, and trigonometry applications sometimes combine with coordinate geometry or 3D figures, so watch for cross-topic question types when revising.
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Using the rules in real situations
The cosine rule and sine rule are not only for exam questions; they are useful in real measurement. For example, to find the angle between two paths, or to work out a third distance from two known distances and an included angle, you can use the cosine rule. The hard part of these application questions is often not the calculation but turning the words into a diagram showing which three pieces are known.
Common application settings include: two sides and the included angle to find the third side (cosine rule), three sides to find an angle (rearranged cosine rule), and two angles and one side to find another side (sine rule). Draw a sketch first, mark all the known sides and angles, then apply the three-step method above.
If the question asks for the area, you can use the area formula area = (1/2)ab sin C, but choose the right pair of sides and their included angle.
Frequently asked questions
If two sides and the angle opposite one of them are given, should I use the sine rule or the cosine rule?
This usually calls for the sine rule, because you already have one complete side-and-opposite-angle pair. Watch for the ambiguous case: the same sine value can belong to an acute and an obtuse angle, so check that the two angles add up to less than 180°, then choose according to the question. Using the cosine rule instead leads to a quadratic equation and is more tedious.
Why can't I use the sine rule when three sides are given to find an angle?
The sine rule needs at least one known angle to set up a usable ratio. With three sides and no known angle, that ratio cannot be written. The rearranged cosine rule cos C = (a² + b² − c²) / (2ab) needs only the three sides, so it is the direct method.
Does a negative cosine value in the cosine rule mean I have made a mistake?
Not necessarily. When the included angle is obtuse, its cosine is negative anyway, so − 2ab cos C becomes a plus. A negative cosine means the angle is greater than 90°, which is a normal result. Only a cos value outside the range −1 to 1 means a calculation error.
When should I consider tutoring?
First work out the nature of the difficulty. If you just did not follow one lesson, rewatching video lessons and doing basic questions usually fills the gap. If you are repeatedly stuck across several topics, or need someone to point out blind spots in your thinking straight away, then tutoring is worth considering. Options include tutoring centres, private tutoring, home tutoring and online tutoring, each with its own trade-offs in time and interaction. Prices vary with format and district and are usually a range, so find out the details before deciding.