Mathematics • Year 9 • Unit 3 • Lesson 7
Sine in the Real World — Slides, Slopes and Kites
Apply sin θ = opp/hyp to water slides, bushwalking trails, kite strings, helicopter winch cables and pedestrian ramps. Every problem involves the OPP + HYP pair — the signature setup for sine.
1. Word problems
Calculator in DEG mode. Quick check: sin 30° must read 0.5. Round to 2 d.p. unless told otherwise.
1.1 — Water slide. A water slide at Wet'n'Wild is 19.5 m long and meets the pool at an angle of 38° to the horizontal. The vertical drop from the top of the slide to the pool is the opposite side.
(a) Find the vertical drop of the slide.
(b) The pool is 1.2 m deep at the catch. How many metres above the pool deck is the top of the slide? 3 marks
1.2 — Bushwalking trail. The Blue Mountains' Furber Steps section averages a 12° slope. A walker covers 200 m along the trail (the slant distance, i.e. the hypotenuse).
(a) Find how many vertical metres the walker has climbed.
(b) If a fitter walker covers 350 m along the same 12° slope, find how many vertical metres they have climbed. 3 marks
1.3 — Kite string. Mei is flying a kite. The string makes a 50° angle with the horizontal. She wants the kite to be exactly 20 m vertically above the ground (treat her hand as ground level).
(a) How long must the kite string be?
(b) If a gust pulls the string angle down to 35° (same string length), what is the new height of the kite? 3 marks
1.4 — Helicopter winch. A rescue helicopter is hovering and lowers a 25 m winch cable to a stranded climber on a cliff face. The cable hangs at 70° to the horizontal (rather than vertical, because of the chopper's downdraft and forward speed).
(a) Find the vertical distance from the helicopter down to the climber (the opp).
(b) If the helicopter rises 5 m, the cable angle becomes steeper. Should the new angle be bigger or smaller than 70°? Explain in one sentence. 3 marks
1.5 — Council pedestrian ramp. A new pedestrian ramp at Town Hall station must rise exactly 1.4 m (from street level up to the platform). Local council guidelines say the ramp angle can be 5° (gentle), 8° (standard) or 10° (steep, only with handrail).
For each angle, find the required length of the ramp (the hypotenuse) and state which option uses the least material. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 Your friend says: "If I need to find the hypotenuse from the opposite, I should MULTIPLY by sin θ, the same way I multiplied by sin θ to find the opposite from the hypotenuse." In your own words, explain (i) why this is wrong, (ii) what the correct operation is, (iii) why dividing by sin θ (a value less than 1 for acute angles) makes the answer bigger than the opp. Refer to the equation sin θ = opp/hyp somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Water slide 19.5 m at 38°
(a) Drop = 19.5 sin 38° ≈ 19.5 × 0.6157 ≈ 12.01 m.
(b) The vertical drop IS the height of the slide top above the pool deck — the pool depth is irrelevant — so ≈ 12.01 m.
Reading carefully matters: not every number in a word problem is needed.
1.2 — Furber Steps 12°
(a) Vertical rise = 200 sin 12° ≈ 200 × 0.2079 ≈ 41.58 m.
(b) Vertical rise = 350 sin 12° ≈ 350 × 0.2079 ≈ 72.77 m.
Same angle, longer distance → proportionally more height. The ratio of heights is 350/200 = 1.75 — try it.
1.3 — Kite at 50° rising to 20 m
(a) hyp = 20 / sin 50° ≈ 20 / 0.7660 ≈ 26.11 m of string.
(b) With the same string length and angle now 35°: new height = 26.11 × sin 35° ≈ 26.11 × 0.5736 ≈ 14.98 m.
So the gust dropped the kite by about 5 m — a real-world effect of wind.
1.4 — Helicopter winch 25 m at 70°
(a) Vertical drop = 25 sin 70° ≈ 25 × 0.9397 ≈ 23.49 m.
(b) Bigger than 70°. The cable length is fixed but the helicopter is higher, so to reach the same climber the cable must hang closer to vertical — i.e. closer to 90° from horizontal.
Sense-check: a perfectly vertical cable would be at 90°, which is the steepest possible.
1.5 — Town Hall pedestrian ramp, rise = 1.4 m
5°: hyp = 1.4 / sin 5° ≈ 1.4 / 0.0872 ≈ 16.06 m.
8°: hyp = 1.4 / sin 8° ≈ 1.4 / 0.1392 ≈ 10.06 m.
10°: hyp = 1.4 / sin 10° ≈ 1.4 / 0.1736 ≈ 8.06 m.
The 10° (steep, with handrail) ramp uses the least material — but it's also the hardest to walk up.
This is exactly the design trade-off accessibility engineers face.
2.1 — Explain your thinking (sample response)
My friend is wrong because the rearrangement direction depends on whether the unknown is on the TOP or the BOTTOM of the fraction. The lesson equation is sin θ = opp/hyp. When the unknown is opp (on top), we multiply both sides by hyp to get opp = hyp × sin θ. But when the unknown is hyp (on the bottom), we have to DIVIDE — first multiply by hyp to get hyp × sin θ = opp, then divide by sin θ to land on hyp = opp / sin θ. So the correct operation is divide, not multiply. For any acute angle θ, sin θ is between 0 and 1, and dividing a positive number by something less than 1 always makes it BIGGER. That's why the hypotenuse comes out larger than the opp — which is exactly what we expect, because the hypotenuse is always the longest side of a right triangle.
Marking: 1 mark for naming "divide"; 1 for explicitly writing hyp = opp/sin θ; 1 for explaining why "dividing by less than 1 makes it bigger"; 1 for connecting back to "hyp is always the longest side".