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S4 Maths: Trigonometric Ratios and Bearings (Paper 3 (challenge))
S4 Maths: Trigonometric Ratios and Bearings (Paper 3 (challenge))
Paper 3 (challenge): ten questions on trigonometric ratios and bearings 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: Trigonometric Ratios and Bearings |
| Level | S4 |
02 Questions (10 items)
- FoundationA tower is 50 m tall and the angle of elevation of its top from a point on the ground is 45 degrees. Find the distance from the point to the foot of the tower.
- FoundationGiven that the angle is acute and its sine is 3 over 5, find the cosine.
- FoundationAn isosceles right-angled triangle has a hypotenuse of 10 root 2 cm. Find the length of each leg.
- IntermediateFrom the foot of a hill the angle of elevation of the summit is 30 degrees. After walking 100 m towards the hill the angle becomes 45 degrees. Find the height of the hill.
- IntermediateA lighthouse lies on a bearing of 060 degrees at a distance of 5 km from a boat. Find the displacements east and north from the boat.
- IntermediateA 13 m ladder leans against a vertical wall with its foot 5 m from the wall. Find the height reached and the angle the ladder makes with the ground.
- ChallengeFrom point A the bearing of point B is 040 degrees. What is the bearing of A from B?
- ChallengeA boat sails 8 km from P on a bearing of 120 degrees to Q, then 6 km on a bearing of 030 degrees to R. Find the total displacement east from P to R.
- ChallengeA kite string 50 m long makes an angle of 60 degrees with the horizontal. Find the height of the kite above the ground.
- Error redoError clinic: given a horizontal distance of 100 m and an angle of elevation of 30 degrees, a student uses cosine to find the height. What is the correct method and answer?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | Distance = 50 / tan 45 = 50 m. | Tangent links height as the opposite side to distance as the adjacent, so finding the distance means dividing. |
| 2 | Cosine is the square root of 1 minus 9 over 25, which is 4 over 5 or 0.8. | The identity sine squared plus cosine squared equals 1 solves this without any side lengths. |
| 3 | Each leg is 1 cm, because the hypotenuse equals a leg times root 2. | In an isosceles right-angled triangle the hypotenuse is always root 2 times a leg. |
| 4 | The height h satisfies h / tan 30 minus h / tan 45 equals 100, giving h of about 136.603 m. | Write one equation for each observation with the same height, and the difference is the 100 m walked. |
| 5 | East = 5 x sin 60, about 4.33 km; north = 5 x cos 60 = 2.5 km. | A bearing starts at north, so the north component uses cosine and the east component uses sine. |
| 6 | Height = square root of (13 squared minus 5 squared) = 12 m; cosine of the angle is 5 over 13, so the angle is about 67.38 degrees. | Pythagoras gives the height, while the angle needs the adjacent over hypotenuse ratio, which is cosine. |
| 7 | Add 180 to 40, giving 22 degrees. | The bearing of one point from the other always differs by 180 degrees. |
| 8 | East displacement = 8 x sin 120 plus 6 x sin 30, about 9.928 km. | Each leg contributes a sine component to the east, and the two components simply add. |
| 9 | Height = 50 x sin 60, about 43.301 m. | The string is the hypotenuse and the height is the opposite side, so sine is the direct choice. |
| 10 | Use tangent instead: height = 100 x tan 30, about 57.735 m. | Finding the opposite side from the adjacent side always calls for tangent, while cosine would only return the hypotenuse. |
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.
For any trigonometric question, start by drawing the triangle, marking the right angle and labelling the three sides: the side facing the angle is the opposite, the side touching the angle beside the right angle is the adjacent, and the longest side is the hypotenuse. The three ratios differ only in the denominator: sine uses the hypotenuse, cosine uses the hypotenuse, and tangent uses the adjacent. Once the ratio is written, see which two quantities are known and pick the formula that contains only those two, leaving a single unknown. Memorise the values for 30, 45 and 60 degrees so no calculator is needed when they appear. Bearings are always measured clockwise from north: 0 is north, 90 is east, 180 is south and 270 is west, so convert an N..E statement by splitting the angle from north or from south and checking that the two parts add up to the bearing given. Finally test the answer against the triangle: the hypotenuse must be the longest side and every ratio must be less than 1.Tutor Registration →
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