Mathematics • Year 9 • Unit 3 • Lesson 6
SOH–CAH–TOA in the Real World
Read trig ratios off real right triangles in skate ramps, roof trusses, phone screens, accessibility ramps and TV brackets — then explain why the ratio of a side pair depends only on the angle, not the size of the triangle.
1. Word problems
You are NOT calculating angles yet — only reading off ratios from given side lengths. Show your working — a final number without working only earns half marks.
1.1 — Skate ramp. A backyard skate ramp has a vertical height (opp) of 1.2 m, a horizontal base (adj) of 1.6 m and a sloped ramp surface (hyp) of 2.0 m. Let θ be the angle the ramp surface makes with the ground.
(a) Calculate sin θ, cos θ and tan θ as decimals.
(b) State which of those three ratios is the steepness ratio (rise over run). 3 marks
1.2 — Roof truss. One half of a symmetrical triangular roof truss is a right triangle with the slanting rafter (hyp) = 5 m, the vertical post (opp) = 3 m, and the horizontal beam from post to wall (adj) = 4 m. Let θ be the pitch angle (between rafter and beam).
(a) Find sin θ and cos θ as fractions.
(b) Check that sin² θ + cos² θ = 1. 3 marks
1.3 — Phone screen diagonal. A rectangular phone screen is 7.2 cm wide and 13.5 cm tall. Drawing the diagonal makes a right triangle: opp = 13.5 cm (the height), adj = 7.2 cm (the width), hyp = 15.3 cm (the diagonal). Let θ be the angle the diagonal makes with the bottom edge.
(a) Find tan θ as a fraction in simplest form (divide top and bottom by 0.9).
(b) State, with reason, whether tan θ is bigger or smaller than 1 in this phone screen. 3 marks
1.4 — Accessibility ramp standard. Australian Standard AS1428 requires that an accessibility ramp must not be steeper than 1:14 (i.e. for every 14 m along the ground, the ramp rises 1 m). For a ramp built EXACTLY at that limit, opp = 1 m and adj = 14 m. Pythagoras gives hyp ≈ 14.036 m.
(a) Find tan θ as a fraction.
(b) Find sin θ to 4 decimal places.
(c) Are tan θ and sin θ almost equal here? Explain why in one sentence. 3 marks
1.5 — TV wall bracket. A TV bracket is a small right triangle of metal with sides 9 cm, 40 cm and 41 cm. Two brackets are made, one for a small TV (the 9–40–41 triangle) and one for a big TV that is a 3× scaling (27–120–123). Let θ be the same acute angle in both brackets.
(a) Find sin θ in the SMALL bracket.
(b) Find sin θ in the BIG bracket.
(c) Compare your answers — what does this tell us about how trig ratios behave when a triangle is scaled? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "Because the big TV bracket is three times the size of the small one, sin θ must be three times bigger too." In your own words, explain (i) why this is wrong, (ii) what stays the same when a right triangle is scaled, and (iii) which lesson concept makes this true. Refer to "same angle → same ratio" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Skate ramp
(a) sin θ = 1.2/2.0 = 0.6; cos θ = 1.6/2.0 = 0.8; tan θ = 1.2/1.6 = 0.75.
(b) Rise over run = vertical over horizontal = opp/adj = tan θ.
This is a 3–4–5 triangle scaled by 0.4 — same angles, same trig ratios.
1.2 — Roof truss (3–4–5)
(a) sin θ = opp/hyp = 3/5 = 0.6; cos θ = adj/hyp = 4/5 = 0.8.
(b) sin² θ + cos² θ = (3/5)² + (4/5)² = 9/25 + 16/25 = 25/25 = 1 ✓.
This identity always works — it's just Pythagoras (opp² + adj² = hyp²) divided through by hyp².
1.3 — Phone screen diagonal
(a) tan θ = 13.5/7.2 = (13.5÷0.9)/(7.2÷0.9) = 15/8 ≈ 1.875.
(b) tan θ is BIGGER than 1, because the height (opp = 13.5) is greater than the width (adj = 7.2), so opp/adj > 1.
A "portrait" phone always has tan(diagonal angle) > 1; a "landscape" tablet would have tan < 1.
1.4 — Accessibility ramp 1:14
(a) tan θ = opp/adj = 1/14 ≈ 0.0714.
(b) sin θ = opp/hyp = 1/14.036 ≈ 0.0713.
(c) Yes — almost identical. When the angle is small the hypotenuse is barely longer than the adjacent (14.036 vs 14.000), so opp/adj ≈ opp/hyp.
For small angles, sin θ ≈ tan θ — a fact engineers use all the time.
1.5 — TV brackets (9–40–41 and 27–120–123)
(a) Small bracket: sin θ = 9/41 ≈ 0.2195.
(b) Big bracket: sin θ = 27/123 = (27÷3)/(123÷3) = 9/41 ≈ 0.2195.
(c) Identical. Scaling a triangle multiplies every side by the same factor (here 3), which cancels in the ratio: (3×9)/(3×41) = 9/41. The angle didn't change, and the ratio didn't either.
This is why your calculator can store ONE number for sin 30° and re-use it forever — the angle is what matters.
2.1 — Explain your thinking (sample response)
My classmate is wrong because scaling a triangle multiplies EVERY side by the same number (here, 3), and that same factor appears on the top AND the bottom of any trig ratio. When you compute sin θ = opp/hyp for the big bracket, you get 27/123, but both numbers share the factor 3, so the ratio simplifies right back to 9/41 — exactly the same value as the small bracket. The lesson concept that makes this true is "same angle → same ratio" — also called the similar-triangles property in geometry. Two right triangles with identical angles are similar, and similar triangles have proportional sides, which means their trig ratios are equal. The bracket got bigger, but the ANGLE θ didn't change — and sin only cares about the angle.
Marking: 1 mark for explicitly saying "scale factor cancels in the ratio"; 1 for "same angle → same ratio" phrase; 1 for naming similar triangles or the L6 concept; 1 for clear full-sentence explanation.