Mathematics • Year 9 • Unit 3 • Lesson 11

Inverse Trig in the Real World

Use sin⁻¹, cos⁻¹ and tan⁻¹ in everyday contexts: skate ramps, ladder safety, mobile phone camera tilt, mountain trails, and a wheelchair access ramp. Then explain your method in your own words.

Apply · Real-World Maths

1. Word problems

Each problem ends with finding an angle, so each one finishes with sin⁻¹, cos⁻¹ or tan⁻¹. Show your working — a single final answer with no working only earns half marks. Round angles to 1 d.p. unless stated.

1.1 — Skate ramp. Pria is building a skate ramp. The vertical kicker is 0.9 m high and the slope (line of the deck) is 1.5 m long.

(a) Identify which of opp, adj, hyp are known relative to the angle the ramp makes with the ground.
(b) Find this angle to 1 d.p.
(c) Skaters say a 30°–40° ramp is "mellow". Would Pria's ramp count as mellow? 3 marks

Stuck? You know opp (0.9 m) and hyp (1.5 m) — use sin⁻¹.

1.2 — Ladder safety. Workplace safety guidance for an extension ladder says the angle to the ground should be 70°–80°. Mei sets up an 8 m ladder with its base 2.5 m from the wall.

(a) Find the angle the ladder makes with the ground (1 d.p.).
(b) Is the angle inside the safe range? If not, suggest whether Mei should move the base CLOSER to or FURTHER from the wall, and roughly by how much, to land at exactly 75°. 4 marks

Stuck on (a)? Relative to the ground angle, you know adj (2.5 m) and hyp (8 m) — use cos⁻¹. For (b), solve cos 75° = base / 8 to find the new base distance.

1.3 — Mobile phone camera tilt. Standing 1.6 m from a wall, you tilt your phone to photograph the top of a 2.4 m painting whose bottom is at your eye level. The line of sight to the top forms a right triangle with the wall.

(a) Identify opp and adj relative to your tilt angle θ above horizontal.
(b) Find θ to 1 d.p. — this is the angle your phone must be tilted up. 3 marks

Stuck? opp = 2.4 m (height above eye level), adj = 1.6 m (your distance from the wall) — use tan⁻¹.

1.4 — Bushwalking trail. A 600 m bushwalking trail climbs steadily and rises 90 m in vertical height from start to finish.

(a) Sketch the trail as a right triangle with the slope as the hypotenuse.
(b) Find the average gradient angle of the trail to 1 d.p.
(c) "Easy" trails are under 5°, "moderate" 5°–10°, "hard" over 10°. Classify this trail. 3 marks

Stuck? opp = 90 m (rise), hyp = 600 m (trail length) — use sin⁻¹.

1.5 — Wheelchair access ramp. Australian standard AS 1428.1 limits wheelchair ramps to a maximum slope of 1 vertical unit for every 14 horizontal units.

(a) Express the slope as tan θ = ____ and find θ to 1 d.p. — the maximum legal ramp angle.
(b) A school's new ramp rises 0.5 m over a horizontal distance of 6 m. Find its angle to 1 d.p. Is it legal? 3 marks

Stuck? Both parts use opp + adj, so use tan⁻¹. Compare the two angles to answer "legal?".

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate writes "sin θ = 0.5 means θ = 1 / 0.5 = 2°". In your own words, explain (i) what mistake they have made, (ii) which idea from Lesson 11 they have confused, and (iii) the correct value of θ. Refer to "inverse function" versus "reciprocal" somewhere in your explanation.

Stuck? Revisit lesson § "Spot the Trap" — sin⁻¹(x) is NOT 1/sin(x).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Skate ramp

(a) opp = 0.9 m (vertical kicker), hyp = 1.5 m (slope). SOH applies.
(b) sin θ = 0.9 / 1.5 = 0.6, so θ = sin⁻¹(0.6) ≈ 36.9°.
(c) Yes — 36.9° sits inside the 30°–40° "mellow" range.
Useful trick: 0.9 : 1.5 = 3 : 5, so this is a "3-4-5" friend angle (sin⁻¹(3/5) ≈ 36.87°).

1.2 — Ladder safety

(a) Relative to the ground angle: adj = 2.5 m, hyp = 8 m. CAH: cos θ = 2.5 / 8 = 0.3125. θ = cos⁻¹(0.3125) ≈ 71.8°.
(b) 71.8° IS inside the safe 70°–80° range — the ladder is set up safely. To hit exactly 75°: cos 75° ≈ 0.2588, so new base = 8 × 0.2588 ≈ 2.07 m. Mei would move the base CLOSER to the wall by about 2.5 − 2.07 ≈ 0.43 m to land at exactly 75°.

1.3 — Phone camera tilt

(a) opp = 2.4 m (height of painting top above eye), adj = 1.6 m (your distance from wall). TOA applies.
(b) tan θ = 2.4 / 1.6 = 1.5. θ = tan⁻¹(1.5) ≈ 56.3° — you'd need to tilt your phone up by about 56°, which is steep enough that the photo will be noticeably distorted (perspective).

1.4 — Bushwalking trail

(a) Right triangle: opp = 90 m, hyp = 600 m, slope along hypotenuse.
(b) sin θ = 90 / 600 = 0.15. θ = sin⁻¹(0.15) ≈ 8.6°.
(c) 8.6° falls inside the 5°–10° range — classify as moderate.

1.5 — Wheelchair access ramp

(a) tan θ = 1 / 14. θ = tan⁻¹(1 / 14) ≈ 4.1° — the maximum legal angle under AS 1428.1.
(b) tan θ = 0.5 / 6 = 1 / 12 ≈ 0.0833. θ = tan⁻¹(1 / 12) ≈ 4.8°. This is steeper than 4.1°, so the ramp would NOT meet the AS 1428.1 standard — the school would need to extend the horizontal run (to at least 7 m for a 0.5 m rise) to make it legal.

2.1 — Explain your thinking (sample response)

My classmate has confused the inverse function with the reciprocal. The symbol sin⁻¹(0.5) does NOT mean 1 / sin(0.5); it means "the angle whose sine is 0.5". These are two completely different operations — the −1 in sin⁻¹ is a notation for "inverse function", not an exponent. The correct method is to enter SHIFT → sin → 0.5 → = on the calculator, which gives θ = 30°. A quick sanity check: sin 30° = 0.5, so 30° really is "the angle whose sine is 0.5", confirming the answer.

Marking: 1 mark for naming the mistake (inverse vs reciprocal); 1 for the explanation in own words; 1 for the correct answer 30°; 1 for the sanity check or worked alternative.