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Sine and Cosine Rules - Edexcel GCSE Higher Maths

Typically 3-6 marks per paper, concentrated on Papers 2 and 3 · Spec 1MA1

The sine rule, the cosine rule and the area formula Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C make up spec reference G22 on Edexcel's 1MA1, and all of it is Higher tier only. Between them they handle any triangle that does not contain a right angle, which is what separates G22 from the SOHCAHTOA ratios at G20. All three formulae are printed on the Higher formulae sheet, so none of this is a memory test. What is being assessed is whether you can look at a triangle, work out which three pieces of information you have, and pick the formula that uses exactly those three.

These marks live on Papers 2 and 3, where you have a calculator. A short question gives one triangle and asks for a side or an angle, usually for 3 marks. A longer one, typically somewhere after question 15, runs to 5 or 6 marks: find a length with the cosine rule and feed it into the sine rule, or find an angle from three sides and then put it into the area formula. Bearings are the standard dressing, and they add a step before any trigonometry starts, because you have to convert the bearings into an angle inside the triangle first. Paper 1 versions do turn up, built on sin⁡30∘=12\sin 30^\circ = \frac{1}{2} and sin⁡45∘=22\sin 45^\circ = \frac{\sqrt{2}}{2}, with the answer left as a surd or a fraction.

The opening mark is for a correct substituted statement, so write the formula out with your numbers in it before you touch the calculator. The fastest way to score nothing is to pair the wrong side with the wrong angle: in asin⁡A\frac{a}{\sin A}, the side aa has to be the one facing angle AA, not one of the two beside it. After that, three slips recur. Forgetting the square root at the end of a cosine rule calculation, so 70.8 goes on the answer line where 8.4 belongs. Reading b2+c2−2bccos⁡Ab^2 + c^2 - 2bc\cos A as (b2+c2−2bc)cos⁡A(b^2 + c^2 - 2bc)\cos A. And handing in the acute angle when the question has already told you the angle is obtuse, which costs an accuracy mark that one subtraction from 180∘180^\circ would have saved. Decide which rule you are using before you calculate anything, because a wrong choice at the start cannot be rescued later.

Worked example

In triangle ABC, AB = 11 cm, BC = 7 cm and AC = 15 cm. Work out the area of triangle ABC. Give your answer in square centimetres correct to 1 decimal place.
[4 marks]
  1. The area formula needs two sides and the angle between them, and no angle is given. Three sides means the cosine rule. Angle ABC sits between AB and BC, so make AC the side that stands alone: cos⁡ABC=112+72−1522×11×7\cos ABC = \frac{11^2 + 7^2 - 15^2}{2 \times 11 \times 7}.
    M1: cosine rule rearranged for the angle, with the 15 cm side isolated because it faces ABC
  2. cos⁡ABC=121+49−225154=−55154=−0.3571…\cos ABC = \frac{121 + 49 - 225}{154} = \frac{-55}{154} = -0.3571\ldots, so angle ABC=110.92…∘ABC = 110.92\ldots^\circ.
    A1: 110.9 to 111.0; the negative cosine is the signal that this angle is obtuse, no adjustment needed
  3. Angle ABC lies between the two sides AB and BC, so it is the included angle the area formula wants: area =12×11×7×sin⁡110.92…∘= \frac{1}{2} \times 11 \times 7 \times \sin 110.92\ldots^\circ.
    M1: correct substitution into half ab sin C, using the two sides that enclose their angle
  4. =38.5×0.9340…=35.96…= 38.5 \times 0.9340\ldots = 35.96\ldots, which is 36.0 square centimetres to 1 decimal place.
    A1: 35.9 to 36.0, from the unrounded angle rather than from 110.9

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams, so every triangle is described in words: sketch it, letter the vertices exactly as the question does, then decide which of the three formulae matches what you have been given before you calculate anything. Nearly all of these are calculator questions, so work in degree mode and hold the full display value between the stages of a two-rule problem.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Exam pace [3 marks]
    A ship leaves port A and sails on a bearing of 070 degrees for 30 km to point B. It then changes course and sails on a bearing of 130 degrees for 40 km to point C. Work out the distance of C from A. Give your answer in km correct to 1 decimal place.
  2. Question 2 Exam pace [3 marks]
    A triangular sail has sides of length 4.2 m, 5.5 m and 7.1 m. Work out the size of the largest angle of the sail. Give your answer in degrees correct to 1 decimal place.
  3. Question 3 Exam pace [3 marks]
    In triangle ABC, angle ABC = 105 degrees, angle BAC = 33 degrees and AB = 7.6 cm. Work out the length of BC. Give your answer in cm correct to 1 decimal place.
  4. Question 4 Exam pace [4 marks]
    In triangle ABC, angle BAC = 50 degrees, angle ABC = 60 degrees and AB = 10 cm. Work out the area of triangle ABC. Give your answer in square centimetres correct to 1 decimal place.
  5. Question 5 Exam pace [3 marks]
    In triangle ABC, angle BAC = 28 degrees, BC = 5.4 cm and AC = 9.2 cm. Angle ABC is obtuse. Work out the size of angle ABC. Give your answer in degrees correct to 1 decimal place.
  6. Question 6 Stretch [5 marks]
    In triangle ABC, AB = 13 cm, AC = 14 cm and BC = 15 cm. Work out the size of angle BAC in degrees correct to 1 decimal place, and the area of triangle ABC in square centimetres correct to 1 decimal place.

Common mistakes examiners see

  • Pairing a side with the angle it sits next to instead of the angle it faces. Given angle ACB, students write ABsin⁡ABC\frac{AB}{\sin ABC} because the letters look similar, when AB is the side that faces angle ACB.

    In three-letter notation the side facing an angle is built from the two letters that are not the middle one, so the side facing ACB is AB. Write the pairs down before substituting: aa with AA, bb with BB, cc with CC. The first mark is for a correctly paired substituted statement, so that line earns something even if the arithmetic then collapses.

  • Stopping at BC2=70.8BC^2 = 70.8 and writing 70.8 cm on the answer line instead of 70.8=8.41\sqrt{70.8} = 8.41 cm.

    Put the squared value on one line and the square root on the next, as two separate steps. Then size-check it: a triangle whose other sides are 9 cm and 7 cm cannot have a 70.8 cm side.

  • Reading the cosine rule as (b2+c2−2bc)cos⁡A(b^2 + c^2 - 2bc)\cos A, so 82+52−2×8×5×cos⁡72∘8^2 + 5^2 - 2 \times 8 \times 5 \times \cos 72^\circ is worked out as 9×cos⁡72∘9 \times \cos 72^\circ and the side comes back as 1.7 cm rather than 8.0 cm.

    Only the 2bccos⁡A2bc\cos A term carries the cosine. Type the whole right-hand side in one go with every multiplication sign in place, and do not press equals partway through.

  • Giving the acute angle when the question has said the angle is obtuse, so a 126.9∘126.9^\circ angle is written down as 53.1∘53.1^\circ. Pressing sin⁡−1\sin^{-1} only ever returns the acute value.

    Scan the stem for the words acute and obtuse before you start. If the angle is obtuse, take your calculator value away from 180∘180^\circ, then check that the three angles of the triangle still total 180.

  • Using the area formula with an angle that is not between the two chosen sides, for example 12×AB×BC×sin⁡BAC\frac{1}{2} \times AB \times BC \times \sin BAC when the angle enclosed by AB and BC is ABC.

    12absin⁡C\frac{1}{2}ab\sin C needs the angle sandwiched between the two lengths you put in. If the angle you have is somewhere else, find the included angle first, either from the 180∘180^\circ angle sum or with the cosine rule, or use a different pair of sides.

Frequently asked questions

Do I get the sine rule and cosine rule in the Edexcel exam?
Yes. Both rules and the area formula, half ab sin C, are printed on the Higher tier formulae sheet that comes with every 1MA1 paper. What you are not given is the decision about which of the three to use, or which side goes with which angle, and that is where almost all of the marks are.
Are the sine and cosine rules on the Foundation paper?
No. Spec reference G22 is Higher tier only, so a Foundation candidate is never asked to work in a triangle without a right angle. Foundation trigonometry stops at the three ratios of G20, which Higher candidates also need.
How do I decide between the sine rule and the cosine rule?
Count what the question gives you. If a side and the angle facing it are both known, the sine rule works. If you have two sides with the angle between them, or all three sides and no angle at all, it is the cosine rule. The mark allocation is a useful hint: 5 or 6 marks usually means one rule and then the other.
What is the ambiguous case of the sine rule?
When you are given two sides and an angle that is not between them, there can be two different triangles that fit, one with an acute angle and one with an obtuse angle. Inverse sine on a calculator only ever reports the acute value, so subtract it from 180 to get the other candidate and check that the angles still add to 180. Edexcel normally settles it for you by stating in the question that the angle is acute or obtuse.

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