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S4 Maths: Sine Rule and Cosine Rule (Paper 3 (challenge))
S4 Maths: Sine Rule and Cosine Rule (Paper 3 (challenge))
Paper 3 (challenge): ten questions on the sine rule and the cosine rule with answers and one-line notes.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S4 Maths: Sine Rule and Cosine Rule |
| Level | S4 |
02 Questions (10 items)
- FoundationA triangle has sides 13 cm, 14 cm and 15 cm. Find its area.
- FoundationIn triangle ABC, B = 110 degrees, C = 25 degrees and b = 20 cm. Find a.
- FoundationA triangle has two sides of 9 cm with an included angle of 60 degrees. Find its area.
- IntermediateA triangle has area 48 cm squared with sides 12 cm and 10 cm. Find all possible included angles.
- IntermediateIn triangle ABC, AB = 10 cm, AC = 7 cm and A = 60 degrees. Find BC.
- IntermediateTwo sides are 11 cm and 8 cm with an included angle of 100 degrees. Find the third side.
- ChallengeIn triangle ABC, a = 7 cm, b = 8 cm and A = 40 degrees. Find all possible B and the number of solutions.
- ChallengeA triangle has sides 4 cm, 5 cm and 6 cm. Find its largest angle.
- ChallengeA person walks 6 km from A, then turns so that the two legs make 120 degrees, and walks 9 km to B. Find the straight distance from A to B.
- Error redoIn triangle ABC, a = 10 cm, b = 12 cm and A = 35 degrees. A student stops at B = 43.495 degrees. How many solutions are there?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | The area is 84 cm squared. | With s = 21 the area is the square root of 21 times 8 times 7 times 6, that is 84. |
| 2 | a = 8.995 cm. | The angle sum gives A = 45 degrees, then a is 20 times sin 45 degrees over sin 110 degrees. |
| 3 | The area is 35.074 cm squared. | Area is 9 times 9 times sin 60 degrees, halved. |
| 4 | C = 53.13 degrees or 126.87 degrees. | sin C is 96 over 120, so both the acute and the obtuse angle work. |
| 5 | BC = 8.888 cm. | BC squared is 100 + 49 – 70, that is 79. |
| 6 | The third side is 14.682 cm. | The cosine of 100 degrees is negative, so the last term adds, giving about 215.6. |
| 7 | B = 47.275 degrees or 132.725 degrees, giving two triangles. | sin B is about 0.735, and 40 plus 132.7 stays under 180, so two triangles fit. |
| 8 | The largest angle is 82.819 degrees. | The largest angle faces the 6 cm side and cos C is 5 over 40. |
| 9 | The distance is 13.077 km. | The cosine rule gives 36 + 81 + 54, that is 171. |
| 10 | B = 43.495 degrees or 136.505 degrees, so there are two solutions. | sin B is about 0.688, and 35 plus 136.5 stays under 180, so both angles are valid. |
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.
Ask what you hold first: three sides, two sides with the included angle, or two angles with a side. Use the sine rule for two angles and a side, and the cosine rule when the included angle is known. When a sine rule gives an angle, check whether 180 degrees minus it is still valid, because that gives a second triangle. For the area, take half the product of two sides times the sine of the included angle.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.
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