Mathematics • Year 9 • Unit 3 • Lesson 8
Cosine in the Real World — Ladders, Wires and Slopes
Apply cos θ = adj/hyp to ladder-safety problems, mobile-phone tower wires, ski-slope horizontal runs, scaffolding bracing and a sliding shop door. Every problem involves the ADJ + HYP pair — the signature setup for cosine.
1. Word problems
Calculator in DEG mode. Round to 2 d.p. unless told otherwise.
1.1 — Ladder safety. A 6 m ladder is leaning against a wall. Workplace safety guidelines recommend the ladder make a 75° angle with the ground (the famous "1 out for every 4 up" rule).
(a) Find the foot-of-ladder distance from the wall.
(b) Verify that this is approximately 1/4 of the ladder length (the "1 out, 4 up" rule). 3 marks
1.2 — Mobile-phone tower guy wire. A guy wire is anchored 12 m from the base of a phone tower and makes a 60° angle with the ground.
(a) Find the length of the wire.
(b) A second wire on the opposite side is anchored 12 m from the base but at 45°. Find its length. 3 marks
1.3 — Ski slope horizontal run. A black-diamond ski run is 400 m long (measured along the slope itself, the hyp) and drops at an average angle of 28° below horizontal.
(a) Find the horizontal distance (adj) the skier covers from the top of the run to the bottom (the "ground footprint").
(b) If the resort wants to fence off a 5 m safety buffer on each side, what's the total horizontal length of fence needed (two sides of a rectangle, just the long edges)? 3 marks
1.4 — Scaffolding cross-brace. A diagonal cross-brace on a scaffolding frame is 3.2 m long and makes a 55° angle with the horizontal bottom rail.
(a) Find the horizontal length the brace spans along the bottom rail.
(b) The horizontal rail is 1.6 m long. Will one brace span it fully, or do you need two cross-braces? 3 marks
1.5 — Sliding shop door. A shop's automatic sliding door track is mounted slightly tilted: the track is 2.5 m long and angled 4° below horizontal so that the door eases shut by gravity.
(a) Find the horizontal distance the door travels when it opens fully.
(b) Find the vertical drop of the door from one end of the track to the other (use sin 4°). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate sees a guy-wire problem and immediately writes "sin θ = adj/hyp" because they always use sine. Explain (i) why this is the wrong ratio for adj+hyp, (ii) which ratio they should use, (iii) what concrete steps they should take BEFORE picking any ratio. Refer to "list the two sides involved" and "CAH" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Ladder safety
(a) Foot = 6 cos 75° ≈ 6 × 0.2588 ≈ 1.55 m.
(b) Foot / ladder = 1.55 / 6 ≈ 0.259 ≈ 1/4. ✓ The 75° rule does give about "1 out for every 4 up" (since cos 75° ≈ 0.26).
This is why ladder safety posters use 75° — it's a memorable rule that maps to simple geometry.
1.2 — Phone tower guy wires
(a) Wire 1 length = 12 / cos 60° = 12 / 0.5 = 24 m exactly.
(b) Wire 2 length = 12 / cos 45° ≈ 12 / 0.7071 ≈ 16.97 m.
The 45° wire is shorter because at 45° the wire travels equal horizontal and vertical distance, vs at 60° most of the wire goes up.
1.3 — Ski run 400 m at 28°
(a) Horizontal distance = 400 cos 28° ≈ 400 × 0.8829 ≈ 353.18 m.
(b) Two long edges of the fence = 2 × 353.18 ≈ 706.35 m.
The slope is 400 m but the ground footprint is shorter — because some of the slope distance "goes downward" instead of along the ground.
1.4 — Scaffolding cross-brace
(a) Horizontal span = 3.2 cos 55° ≈ 3.2 × 0.5736 ≈ 1.84 m.
(b) Since 1.84 m > 1.6 m, one brace is enough to span the 1.6 m rail (with about 24 cm to spare in horizontal projection).
1.5 — Sliding shop door, 2.5 m track at 4°
(a) Horizontal distance = 2.5 cos 4° ≈ 2.5 × 0.9976 ≈ 2.49 m (almost the full track — 4° is very shallow).
(b) Vertical drop = 2.5 sin 4° ≈ 2.5 × 0.0698 ≈ 0.17 m (just 17 cm — enough for gravity to ease the door closed).
Real engineering decision: 4° is small enough to be invisible to customers but enough to slowly close the door.
2.1 — Explain your thinking (sample response)
My classmate is wrong because sin θ is defined as opp/hyp, not adj/hyp. The two sides in this guy-wire problem are the adjacent (ground distance) and the hypotenuse (the wire), so the correct ratio is CAH — cosine = adj/hyp. The proper steps to take BEFORE picking any ratio are: (1) sketch the triangle and mark θ; (2) list the two sides involved in the question (one known + one wanted); (3) match that pair to the correct ratio — opp+hyp → sin, adj+hyp → cos, opp+adj → tan; and (4) only THEN write the equation and rearrange. Picking a ratio by habit instead of by the side-pair is the most common Year 9 mistake, and it always leads to wrong numbers, because each ratio is locked to a specific pair of sides.
Marking: 1 mark for "sin uses opp/hyp"; 1 mark for naming CAH = cos as the correct ratio; 1 mark for the phrase "list the two sides involved"; 1 mark for clear full-sentence explanation referring to side-pairs.