Why do error drills work better than fresh questions?

The slips usually sit in the same one or two steps, so retrying the same setting shows at once whether they are fixed.

When do I use the sine rule rather than the cosine rule?

Use the sine rule for two angles with a side or two sides with a non-included angle; use the cosine rule for two sides with the included angle or for three sides.

How do I spot the ambiguous case?

When reversing the sine rule, if b sinA is less than a and b is longer than a, both the acute and obtuse values must be considered.

Where do bearing questions lose marks?

Almost always on the included angle: bearings are subtracted directly, so draw the north arrows before reaching for 180 degrees.

S4 Maths sine and cosine rule paper 4: ten common slips

2026-11-10

Tutor Prep Guide · S4 Maths Sine and Cosine Rule

Ten items rewritten from the usual slips: inverted sines, a wrong cosine sign, the two-solution test and bearing angles.

Direct answers　Items 01 to 03 retry the inverted sine ratio, subtracting a negative cosine and the area formula, 04 to 06 reverse angles, two solutions and Heron, 07 to 09 bearings, elevation and a parallelogram, and 10 the numerator order in the cosine rule.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Sine and Cosine Rule
Level	S4

02　Questions (10 items)

FoundationIn triangle ABC, a = 8 cm, A = 40 degrees and B = 60 degrees. Find b. A common slip is writing b = 8 sin40 / sin60.
FoundationTwo sides of 6 cm and 9 cm include an angle of 110 degrees. Find the third side c.
FoundationTwo sides of 7 cm and 10 cm include an angle of 55 degrees. Find the area.
IntermediateA triangle has sides 7 cm, 8 cm and 9 cm. Find the largest angle.
IntermediateGiven a = 9 cm, A = 35 degrees and b = 12 cm, find all possible values of B.
IntermediateA triangle has sides 11 cm, 13 cm and 20 cm. Find its area using Heron formula.
ChallengeFrom P, Q lies 6 km away on a bearing of 060 and R lies 8 km away on a bearing of 150. Find QR.
ChallengeFrom A the elevation of a tower top is 28 degrees; after walking 60 m nearer the elevation is 45 degrees. Find the height of the tower.
ChallengeA parallelogram has adjacent sides 8 cm and 11 cm with an included angle of 65 degrees. Find its longer diagonal.
Error redoA student finds the largest angle of a 7-8-9 triangle as cosC = (81 – 49 – 64) / (2 x 7 x 8) = -0.286, giving 106.6 degrees. What went wrong?

03　Answers and one-line notes

#	Answer	One-line note
1	10.8 cm	By the sine rule a / sinA = b / sinB, so b = a sinB / sinA = 8 sin60 / sin40 = 10.8 cm; inverting the sines is the usual first mistake.
2	12.4 cm	c^2 = a^2 + b^2 – 2ab cosC = 36 + 81 – 2 x 6 x 9 x cos110 = 153.9, so c = 12.4 cm; cos110 is negative, and subtracting a negative adds.
3	28.7 cm^2	Area = 0.5 x 7 x 10 x sin55 = 28.7 cm^2; using cos instead of sin is the usual slip.
4	73.4 degrees	The largest angle faces the longest side, cosC = (49 + 64 – 81) / (2 x 7 x 8) = 0.286, so C = 73.4 degrees; a wrong order in the numerator gives a negative value.
5	49.9 or 130.1 degrees	sinB = 12 sin35 / 9 = 0.765, so B = 49.9 or 130.1 degrees; since b sinA = 6.88 cm is less than a = 9 cm, both solutions are valid.
6	66.0 cm^2	s = (11 + 13 + 20) / 2 = 22, so area = sqrt(22 x 11 x 9 x 2) = sqrt(4356) = 66.0 cm^2; forgetting the square root gives 4356.
7	10.0 km	The bearings differ by 150 – 60 = 90 degrees, so QR = sqrt(36 + 64) = 10.0 km; subtracting a bearing from 180 gives the wrong included angle.
8	68.1 m	With height h, h / tan28 – h / tan45 = 60, so h(1.881 – 1) = 60 and h = 68.1 m; forgetting to subtract mixes the two distances up with the height.
9	16.1 cm	The longer diagonal satisfies d^2 = 64 + 121 + 2 x 8 x 11 x cos65 = 259.4, so d = 16.1 cm; only the shorter diagonal uses a minus sign.
10	The numerator is in the wrong order: the square of the opposite side is the one subtracted, so 49 + 64 – 81 = 32 and the answer is 73.4 degrees	cosC = (a^2 + b^2 – c^2) / 2ab, and the opposite side is the one subtracted, so the numerator is 49 + 64 – 81 = 32 and C = 73.4 degrees; a misplaced sign turns an acute angle into an obtuse one.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

Book a Maths tutor Tutor Registration →

Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.