4 high-yield, syllabus-aligned questions on Bearings & trigonometry, covering 12 individually creditable mark points. Each one shows the answer that scores full marks under our guide, which phrase earns which mark, and a common incomplete answer — so you can see the difference rather than guess at it.
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Every question and solution is reviewed by examiners versed in WAEC and JAMB, each with more than thirty years of experience.
Mark allocations are TopMarks’ own, written against the published syllabus and Chief Examiners’ reports. They are a guide to how these answers are usually credited, not an official marking scheme.
Calculate4 marks
A man walks 6 km due north and then 8 km due east. Calculate (a) his distance from the starting point (b) his bearing from the starting point, to the nearest degree.
TopMarks model answer
The two legs are at right angles, so by Pythagoras the distance = √(6² + 8²) = √(36 + 64) = √100 = 10 km. Let θ be the angle of the final position east of north. tan θ = 8 ÷ 6 = 1.3333, so θ = 53°. Bearings are measured clockwise from north, so the bearing is 053°.
Suggested marking guide
1Uses Pythagoras
2Correct distance with the unit
3Correct trigonometric ratio for the angle
4Correct bearing in three figures
The working, line by line
1Sketch the journey: 6 km north, then 8 km east — a right-angled triangle — Always draw the diagram. It shows you which side is opposite and which is adjacent, and it is often worth a mark by itself.
2Distance = √(6² + 8²) = √100 = 10 km — The hypotenuse is the direct distance from start to finish.
3tan θ = opposite ÷ adjacent = 8 ÷ 6, so θ = 53° — Measure the angle FROM the north line, because that is what a bearing is measured from.
4Bearing = 053° — Bearings are always written with THREE figures, clockwise from north. 53° is not an acceptable form.
Why it scores
The three-figure convention is a mark. A bearing of 53° must be written 053°, and one of 7° must be written 007°. It is the cheapest mark in the topic and it is lost every year.
A common incomplete answer
“Distance = √(36 + 64) = 10 km”
What it costs: Two marks of four. Part (a) is fully correct and credited, but the bearing — the second half of the question — was never calculated.
Key ideas to include: Pythagoras, bearing, clockwise from north, three-figure bearing, tan θ.
Calculate3 marks
From a point on level ground 50 m from the foot of a tower, the angle of elevation of the top of the tower is 32°. Calculate the height of the tower, correct to two significant figures. [tan 32° = 0.6249]
TopMarks model answer
Let h be the height of the tower. The height is opposite the angle of elevation and the 50 m is adjacent to it, so tan 32° = opposite ÷ adjacent = h ÷ 50. Therefore h = 50 x tan 32° = 50 x 0.6249 = 31.245. Correct to two significant figures, the height of the tower is 31 m.
Suggested marking guide
1Identifies the correct ratio
2Correct substitution
3Correct answer to 2 s.f. with the unit
The working, line by line
1Sketch: a right-angled triangle with the 32° angle at the ground, height h opposite, 50 m adjacent — Label the sides opposite and adjacent RELATIVE TO THE GIVEN ANGLE. This is what tells you to use tan.
2tan 32° = h ÷ 50 — Opposite over adjacent is tan. Choosing sin or cos here is choosing the wrong two sides.
3h = 50 x 0.6249 = 31.245 m — Multiply, do not divide — h is on the top of the fraction.
4To 2 s.f., h = 31 m — The question asked for two significant figures. Leaving 31.245 loses the accuracy mark.
Why it scores
Two decisions decide this question and both are made before any arithmetic: which sides are opposite and adjacent to the given angle, and therefore which ratio to use. Sketching and labelling the triangle makes both automatic — and the sketch itself often carries the method mark.
A common incomplete answer
“h = 50 x 0.6249 = 31.245 m”
What it costs: One mark of three at least. The substitution and value are credited, but the ratio is never identified, and the answer is not given to the two significant figures the question demanded.
Key ideas to include: angle of elevation, opposite, adjacent, tangent ratio, significant figures.
Explain3 marks
Explain the difference between the bearing of B from A and the bearing of A from B, and find the bearing of A from B if the bearing of B from A is 072°.
TopMarks model answer
A bearing is always measured clockwise from the north direction at the point you are measuring FROM. The bearing of B from A is therefore measured at A, and the bearing of A from B is measured at the north line at B; the second is called the back bearing of the first. Since 072° is less than 180°, the back bearing is 072° + 180° = 252°.
Suggested marking guide
1A bearing is measured clockwise from north at the starting point
2The reverse is the back bearing, differing by 180°
3Correct answer
The working, line by line
1Bearing of B from A = 072° (measured at A) — The point named SECOND is where you stand. "B from A" means you are at A.
2Back bearing: add 180° if the bearing is less than 180°, subtract 180° if it is more — This keeps the answer in the range 000° to 360°.
3072° + 180° = 252° — Write it as three figures.
Why it scores
The phrase “B from A” tells you where to stand: at A. Reading it backwards is the single biggest source of error in bearings, and it produces an answer 180° out — which is exactly the wrong answer the mark scheme is watching for.
A common incomplete answer
“The bearing of A from B is 252°.”
What it costs: Two marks of three. The answer is correct, but the question also asked for the difference to be EXPLAINED, and no explanation is given.
Key ideas to include: bearing, back bearing, clockwise from north, 180° rule.
Find2 marks
Given that sin θ = 3/5 and θ is acute, find the value of tan θ.
TopMarks model answer
sin θ = opposite ÷ hypotenuse = 3/5, so in the right-angled triangle the opposite side is 3 and the hypotenuse is 5. By Pythagoras the adjacent side = √(5² - 3²) = √(25 - 9) = √16 = 4. Therefore tan θ = opposite ÷ adjacent = 3/4.
Suggested marking guide
1Finds the adjacent side by Pythagoras
2Correct value of tan θ
The working, line by line
1Draw a right-angled triangle with opposite = 3 and hypotenuse = 5 — The sine ratio tells you two of the three sides. Draw them.
2Adjacent = √(5² - 3²) = √16 = 4 — Pythagoras. Subtract, because you are finding a shorter side from the hypotenuse.
3tan θ = 3/4 — Leave it as an exact fraction unless a decimal is asked for.
Why it scores
You are never asked to find the angle here, and you should not try. Sketching the 3-4-5 triangle gives every ratio at once, exactly, and without tables. Reaching for a calculator to find θ first introduces rounding error into an answer that should be exact.
A common incomplete answer
“tan θ = 3/4”
What it costs: One mark of two. The value is correct, but the adjacent side is never found, so the method mark cannot be awarded.
Key ideas to include: sine ratio, opposite, hypotenuse, adjacent, Pythagoras, exact value.
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