Mathematics • Year 9 • Unit 3 • Lesson 9
Tangent Ratio — Mixed Challenge
Push your tangent skills further: pick the right rearrangement, combine with eye-height adjustments, use tan⁻¹ to recover an angle, spot a mistake in someone's working, and design a building-height measurement experiment.
1. Mixed problems — pick the right tan setup
Decide whether to MULTIPLY or DIVIDE by tan θ before you start. 3 marks each
1.1 A right triangle has adj = 32 and θ = 27°. Find opp.
1.2 A right triangle has opp = 16 and θ = 42°. Find adj.
1.3 A boat at sea sees a 90 m cliff at an angle of elevation of 5°. Find the horizontal distance from the boat to the cliff base.
1.4 Two surveyors stand at points A and B, 50 m apart on level ground, both on the same side of a building. From A, the angle of elevation to the top of the building is 30°; from B (which is between A and the building) it is 60°. Find the height of the building and the distance from B to the building base. (Set up two tan equations: from A, height = (50 + x) tan 30°; from B, height = x tan 60°. Solve.)
1.5 Using only tan and Pythagoras (no sin or cos), find the hypotenuse of a right triangle with adj = 10 and θ = 35°. (First find opp using tan, then use Pythagoras.)
1.6 A surveyor measures opp = 15 m (tree height above eye level) and adj = 20 m (her distance from the tree). Use tan⁻¹ to find the angle of elevation θ to 1 d.p. Then check: does 15 × cos θ / sin θ recover 20?
2. Find the mistake
A student tries to find the height of a building given a ground distance and an angle of elevation. Exactly one line contains a Year 9 trig-choice mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: Standing 40 m from a building, the angle of elevation to the top is 50°. Find the building height (above eye level).
Student's working:
Line 1: adj = 40, θ = 50°, opp = ?
Line 2: sin 50° = opp / 40
Line 3: opp = 40 × sin 50°
Line 4: opp ≈ 40 × 0.7660 ≈ 30.64 m
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong — focus on the SIDE-PAIR confusion, not arithmetic.
(c) Write out the corrected working from that line onward, including the corrected final answer.
Stuck? The 40 m is the ADJACENT (ground distance), not the hypotenuse. With adj + opp involved (no hyp), the right ratio is TAN.3. Open-ended challenge — measure your school building
This question has more than one valid answer. 4 marks
3.1 You want to estimate the height of your school's tallest building using only a tape measure and a phone protractor app (which measures angles of elevation). Design a measurement plan and present a worked sample using realistic numbers.
Your design must include:
(i) Choose a horizontal distance d (adj) from the building (between 10 m and 50 m).
(ii) Choose a plausible angle of elevation θ (between 20° and 60°) you might measure.
(iii) Calculate opp (height above eye level) = d × tan θ.
(iv) Add a realistic eye-height (say 1.55 m for a Year 9 student) to give the total building height.
(v) Briefly state one source of measurement error and how you would minimise it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — adj = 32, θ = 27°
opp = 32 tan 27° ≈ 32 × 0.5095 ≈ 16.30.
1.2 — opp = 16, θ = 42°
adj = 16 / tan 42° ≈ 16 / 0.9004 ≈ 17.77.
1.3 — Boat at sea, 90 m cliff, 5°
adj = 90 / tan 5° ≈ 90 / 0.0875 ≈ 1028.74 m (just over 1 km offshore — that's why ships rely on lighthouses).
1.4 — Two surveyors
From A: height = (50 + x) tan 30°. From B: height = x tan 60°.
Set equal: x tan 60° = (50 + x) tan 30°.
x (1.7321) = (50 + x)(0.5774).
1.7321 x = 28.87 + 0.5774 x.
1.1547 x = 28.87, so x ≈ 25.00 m.
Height = 25 × tan 60° ≈ 25 × 1.7321 ≈ 43.30 m.
1.5 — adj = 10, θ = 35°, find hyp via tan + Pythagoras
opp = 10 tan 35° ≈ 10 × 0.7002 ≈ 7.00.
hyp = √(10² + 7.00²) ≈ √(100 + 49) ≈ √149 ≈ 12.21.
Cross-check with cos: hyp = 10 / cos 35° ≈ 10 / 0.8192 ≈ 12.21 ✓.
1.6 — opp = 15, adj = 20, find θ
tan θ = 15/20 = 0.75. θ = tan⁻¹(0.75) ≈ 36.9°.
Check: 15 × cos(36.9°)/sin(36.9°) ≈ 15 × (0.7997/0.6004) ≈ 15 × 1.3322 ≈ 19.98 ≈ 20 ✓.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student used sin θ = opp/40, but 40 is the ADJACENT (ground distance), not the hypotenuse. Sine is opp/HYP. With opp + adj involved and no hyp, the correct ratio is tan (TOA), not sin. Picking the wrong ratio because of the wrong side-name is the classic Year 9 mistake the lesson's "Spot the Trap" card warns about.
(c) Corrected from Line 2:
tan 50° = opp / 40
opp = 40 × tan 50°
≈ 40 × 1.1918
≈ 47.67 m.
Sense-check: at 50° (steeper than 45°), height should be bigger than the ground distance. 47.67 > 40 ✓ — the corrected answer matches that prediction; the wrong sin answer of 30.64 didn't.
3 — School building measurement (sample solution)
Sample design:
(i) Stand d = 30 m from the building (measured with tape).
(ii) Hold the phone level and tilt up to the top corner. Read θ = 35° on the protractor app.
(iii) opp = 30 × tan 35° ≈ 30 × 0.7002 ≈ 21.01 m (height above eye level).
(iv) Add eye height: total height ≈ 21.01 + 1.55 = 22.56 m (roughly a 7-storey building).
(v) Main error source: phone tilt — even 2° error in θ causes about 1 m error in opp. Minimise by using a tripod or steadying the phone against a stable wall, and by taking three readings and averaging.
Marking: 1 mark for parts (i)–(iii) computed correctly; 1 mark for adding eye height in (iv); 1 mark for naming a realistic error source in (v); 1 mark for a sensible minimisation strategy.