To learn the Sine and Cosine Rules quickly and correctly, begin by looking at the information given about a triangle. For a right-angled triangle, you may already know Pythagoras and SOH CAH TOA. But many triangles do not contain a 90° angle, and the sine rule and cosine rule let us solve them accurately. The most important step is to match each side with its opposite angle and decide which relationship fits the known information. These eduKateSG Mathematics Quick Notes teach the choice, the calculation and the check with simple worked examples.
- Label carefully: side a is opposite angle A, b opposite B and c opposite C.
- Sine rule: a/sin A = b/sin B = c/sin C.
- Cosine rule: c² = a² + b² − 2ab cos C, with C between sides a and b.
- Triangle angle sum: A + B + C = 180°.
- Area using two sides and included angle: area = ½ab sin C.
- Check: choose degree mode for degree angles and verify the triangle is geometrically possible.
First, learn the letters rather than treating the formulas as magic. Draw triangle ABC and label side a opposite angle A, side b opposite angle B and side c opposite angle C. That correspondence never changes, even if the triangle is rotated. If side a measures 5 cm and angle A is 30°, those two pieces of information form a matching pair. The sine rule connects such pairs. Mark the known side-angle pairs on the picture before substituting into any formula; this prevents swapping an opposite angle with an adjacent one.
Second, use the sine rule when you know a side and its opposite angle plus enough information to find another matching pair. For example, let A = 30°, B = 45° and side a = 6 cm. The sine rule gives b/sin 45° = 6/sin 30°, so b = 6 sin 45° / sin 30° = 6√2 cm ≈ 8.49 cm. The third angle is 180° − 30° − 45° = 105°. Notice that the side opposite the larger 45° angle is longer than the side opposite the 30° angle, as it should be.
Third, use the cosine rule when you know two sides and their included angle, or all three sides and want to find an angle. Suppose a = 5 cm, b = 7 cm and included angle C = 60°. The opposite side c satisfies c² = 5² + 7² − 2(5)(7) cos 60° = 25 + 49 − 35 = 39. Therefore c = √39 cm ≈ 6.24 cm to three significant figures. This equation generalises Pythagoras: if C were 90°, cos 90° = 0, so the subtraction term would disappear and c² would equal a² + b².
Worked example: find a missing side with the sine rule
Triangle ABC has A = 40°, B = 70° and a = 10 cm. Find side b. Label b opposite B and a opposite A. Use b/sin 70° = 10/sin 40°. Therefore b = 10 sin 70° / sin 40° ≈ 14.6 cm to three significant figures. Check the direction: 70° is larger than 40°, so its opposite side should be longer. The approximate value 14.6 cm is indeed greater than 10 cm.
Worked example: find an angle with three given sides
A triangle has sides a = 5 cm, b = 6 cm and c = 7 cm. Find angle C opposite the 7 cm side. Rearrange the cosine rule: cos C = (a² + b² − c²)/(2ab). Substitute: (25 + 36 − 49)/(2 × 5 × 6) = 12/60 = 0.2. Hence C = cos⁻¹(0.2) ≈ 78.5° to one decimal place. Since c is the longest side, C should be the largest angle; the result is consistent with the triangle’s shape.
What information tells you which rule to use?
- Two angles and one side: find the third angle, then usually use the sine rule.
- Two sides and a known non-included opposite angle: the sine rule may apply, but check for an ambiguous case.
- Two sides with the included angle: use the cosine rule to find the third side.
- Three sides: use the cosine rule to find a required angle.
- Two sides with their included angle and area required: use ½ab sin C.
- A right triangle: Pythagoras or basic trigonometric ratios may be quicker.
Worked example: area with two sides and the included angle
Two sides of a triangle measure 8 cm and 5 cm, and the angle between them is 30°. The perpendicular height relative to the 8 cm base is 5 sin 30° = 2.5 cm. Therefore area = ½ × 8 × 2.5 = 10 cm². The compact equivalent formula is ½ × 8 × 5 × sin 30° = 10 cm². This formula works whether or not the triangle is right-angled, provided the chosen angle is the included angle between the two named sides.
Why the cosine rule contains subtraction
With a fixed pair of sides a and b, changing the included angle C changes the distance between their free endpoints. At C = 90°, the cosine term is zero, giving Pythagoras. At an acute angle, cos C is positive, so c² is less than a² + b². At an obtuse angle, cos C is negative, so the subtraction of a negative quantity makes c² greater than a² + b². This geometric direction check can help you spot a wrong sign in the cosine rule.
Be careful with the ambiguous sine-rule case
If you know two sides and an angle that is not included between them, solving with sine may give two possible triangles. For example, let A = 30°, a = 5 cm and b = 8 cm. The sine rule gives sin B = (8 sin 30°)/5 = 0.8. One answer is B ≈ 53.1°, but another angle with the same sine is 180° − 53.1° ≈ 126.9°. Both can form a triangle with A = 30° because each leaves a positive third angle. Do not accept only the calculator’s principal inverse-sine result without checking the possible supplementary angle and the triangle sum.
The ambiguous case can also give one or no triangle
The sine rule does not guarantee two answers. If the second possible angle makes the triangle sum reach or exceed 180°, reject it. If the calculated sine of an angle is greater than 1 or less than −1, the given measurements do not form a real triangle under that arrangement. A right-angle boundary can also yield one valid triangle. Always examine the diagram, compare the possible angle sizes and verify that all three angles are positive and total 180°.
How to use a calculator correctly
- Check whether the problem states degrees or radians.
- For questions with °, select degree mode.
- Write the formula before entering values.
- Keep brackets around numerators, denominators and the entire inverse-trig input.
- Carry sufficient precision through intermediate calculations.
- Round the final length or angle only to the precision requested.
- Check the result against the side-angle order and total triangle angle sum.
Formula guide with side-angle matching
- Side-angle pairs: a opposite A, b opposite B, c opposite C.
- Sine rule: a/sin A = b/sin B = c/sin C.
- Cosine rule: a² = b² + c² − 2bc cos A.
- Equivalent cosine form: b² = a² + c² − 2ac cos B.
- Equivalent cosine form: c² = a² + b² − 2ab cos C.
- Area: ½bc sin A = ½ac sin B = ½ab sin C.
Common mistakes and their repair
- Matching a side with an adjacent angle: each lowercase side belongs to its opposite uppercase angle.
- Using the sine rule with no known matching pair: a cosine-rule start may be necessary.
- Using the wrong included angle: confirm it sits between the two known sides in the cosine formula.
- Forgetting the minus sign: cosine rule uses a² + b² − 2ab cos C.
- Ignoring a second inverse-sine angle: test the ambiguous case.
- Reporting an impossible triangle: verify positive angles summing to 180° and plausible side lengths.
- Using radians for a degree question: check the calculator mode.
Mini-test: choose the rule and solve
- Triangle ABC has A = 30°, B = 45° and a = 6 cm. Find b in exact form.
- A triangle has sides 5 cm and 7 cm enclosing a 60° angle. Find the third side in exact form.
- Find the area of a triangle with two sides 8 cm and 5 cm and included angle 30°.
- A triangle has side lengths 5 cm, 6 cm and 7 cm. Find the angle opposite 7 cm to one decimal place.
- Does A = 30°, a = 5 cm, b = 8 cm allow more than one triangle? Explain briefly.
Answers with the correct reasoning
1. b = 6 sin 45° / sin 30° = 6√2 cm. 2. c² = 25 + 49 − 70 cos 60° = 39, so c = √39 cm. 3. Area = ½ × 8 × 5 × ½ = 10 cm². 4. cos C = (25 + 36 − 49)/60 = 0.2, giving C ≈ 78.5°. 5. Yes, because sin B = 0.8 allows B ≈ 53.1° or 126.9°, and both leave a valid third angle.
Your twelve-minute sine and cosine rule routine
- Minutes 1–2: draw and label A, B, C and their opposite sides a, b, c.
- Minutes 3–5: solve a sine-rule question with a known opposite side-angle pair.
- Minutes 6–8: solve a cosine-rule question with two sides and their included angle.
- Minutes 9–10: calculate an area with ½ab sin C.
- Minutes 11–12: test one ambiguous sine-rule example and check the triangle angle sums.
The aim is to choose the rule because of the information shown, not because one formula is more familiar. When sides match their opposite angles, the included angle is correctly identified and the result is checked geometrically, non-right triangle questions become much easier.
Continue with connected Mathematics Quick Notes
Use these lessons for the next step: Trigonometric Ratios · Pythagoras’ Theorem · Angles · Similarity and Congruence. Test yourself on a changed question and check the result before consulting worked answers.
Read the broader teaching approach in Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original reference remains unchanged.