When do I use the sine rule?

Use a / sinA = b / sinB = c / sinC when two angles and any side are known, or two sides and an angle opposite one of them.

When do I use the cosine rule?

Use a^2 = b^2 + c^2 - 2bc cosA when two sides and the included angle are known, or when all three sides are known and an angle is wanted.

Do obtuse angles need special care?

Yes. The cosine is negative between 90 and 180 degrees, so the subtracted term becomes positive and the answer can exceed both given sides.

Which formula pairs well with the area rule?

Area = 0.5 ab sinC works well with the cosine rule: find the side first, or work backwards from the area to the angle.

S4 Maths sine and cosine rule paper 1: basic substitution

2026-11-10

Tutor Prep Guide · S4 Maths Sine and Cosine Rule

Ten items on basic substitution, starting by matching each side with its opposite angle.

Direct answers　Items 01 to 03 cover the sine rule, 04 to 06 the cosine and area rules, 07 to 09 obtuse angles, reverse angles and the largest angle, and 10 an inverted ratio in the sine 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 = 40, B = 60 and a = 8 cm. Find b.
FoundationIn a triangle B = 30, C = 100 and b = 5 cm. Find c.
FoundationIn a triangle a = 7 cm, A = 50 and b = 9 cm. Find B.
IntermediateIn a triangle b = 5 cm, c = 7 cm and A = 60. Find a.
IntermediateIn a triangle a = 8 cm, b = 6 cm and C = 90. Find c.
IntermediateIn a triangle a = 6 cm, b = 8 cm and C = 30. Find the area.
ChallengeIn a triangle a = 5 cm, b = 7 cm and C = 120. Find c.
ChallengeTwo sides of a triangle are 6 cm and 10 cm and the area is 15 cm². Find the included angle.
ChallengeThe three sides of a triangle are 5 cm, 6 cm and 7 cm. Find the largest angle.
Error redoA student writes b = a sinA / sinB and gets 8 x sin40 / sin60 = 5.94 cm. What is wrong, and what is the correct b?

03　Answers and one-line notes

#	Answer	One-line note
1	10.8 cm	The sine rule gives a / sinA = b / sinB, so b = 8 x sin60 / sin40 = 10.8 cm; each side is paired with its own opposite angle.
2	9.85 cm	c = b sinC / sinB = 5 x sin100 / sin30 = 9.85 cm, with 100 degrees opposite c and 30 degrees opposite b.
3	80.0 degrees	sinB = b sinA / a = 9 x sin50 / 7 = 0.985, so B = 80.0 degrees.
4	6.24 cm	The cosine rule gives a^2 = b^2 + c^2 – 2bc cosA = 25 + 49 – 35 = 39, so a = 6.24 cm; the middle term is subtracted.
5	10.0 cm	cos90 is zero, so the cosine rule reduces to Pythagoras and c = sqrt(64 + 36) = 10.0 cm.
6	12 cm²	Area = 0.5 ab sinC = 0.5 x 6 x 8 x sin30 = 12 cm²; the sine of the included angle must not be dropped.
7	10.4 cm	c^2 = 25 + 49 – 2 x 35 x cos120 = 74 + 35 = 109, so c = 10.4 cm; cos120 is negative, so the subtraction turns into an addition.
8	30 or 150 degrees	sinC = 2 x 15 / (6 x 10) = 0.5, so C can be 30 or 150 degrees and both give the same area.
9	78.5 degrees	The largest angle faces the longest side: cosC = (25 + 36 – 49) / (2 x 5 x 6) = 0.2, so C = 78.5 degrees.
10	The ratio is inverted; the correct b is 10.8 cm	Cross multiplying a / sinA = b / sinB gives b = a sinB / sinA = 8 x sin60 / sin40 = 10.8 cm.

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.

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