Mathematics • Year 9 • Unit 3 • Lesson 7
Sine Ratio, Mixed Challenge
Push your sine skills further: pick the right rearrangement, combine with Pythagoras, spot a mistake in someone's working, and design a two-stage slope problem. Decimal mode (DEG) throughout.
1. Mixed problems, pick the right sine setup
Decide whether to MULTIPLY or DIVIDE by sin θ before you start. Round to 2 d.p. unless told. 3 marks each
1.1 A right triangle has hyp = 22, θ = 42°. Find opp and then find adj (use Pythagoras).
1.2 A right triangle has opp = 8.4, θ = 65°. Find hyp.
1.3 A flagpole's shadow makes a right triangle with the pole. The sun's angle of elevation is 32°. The pole is 12 m tall (opp). Find the length of the cable that would be stretched from the sun-side tip of the shadow to the top of the pole (the hyp).
1.4 Compare ramps. Ramp X is 10 m long at 18°. Ramp Y is 10 m long at 36° (double the angle). Find the vertical rise of each (2 d.p.) and state by what factor the rise increased. Is it exactly 2? Explain in one sentence.
1.5 A surveyor measures the slope of a hillside and finds that walking 50 m up the slope lifts you 8 m vertically. Find the slope angle θ in degrees by first finding sin θ and then using your calculator's sin⁻¹ key (round θ to the nearest degree).
1.6 A 6 m water slide is built at 30°. A second section is added on directly below it that is also straight and drops a further 2 m vertically at 60°. Find the total slant length of the two sections combined (2 d.p.).
2. Find the mistake
A student has tried to find the length of a slide given the vertical drop and the slide angle. Exactly one line contains a Year 9 sine-rearrangement mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A water slide drops 9 m vertically at an angle of 40°. Find the slide length.
Student's working:
Line 1: opp = 9 m, θ = 40°, hyp = ?
Line 2: sin 40° = opp / hyp = 9 / hyp
Line 3: hyp = 9 × sin 40°
Line 4: hyp ≈ 9 × 0.6428 ≈ 5.79 m
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong, and what should have happened.
(c) Write out the corrected working in full from that line onward, including the corrected final answer.
3. Open-ended challenge, your design brief
This question has more than one valid answer. 4 marks
3.1 For this simplified maths model, use a maximum gradient of 1:14 (vertical rise : horizontal run) for a wheelchair-access ramp. For the straight sloped section, this means θ ≤ tan⁻¹(1/14) ≈ 4.09°. The model considers only this gradient limit and a maximum sloped length of 18 m; it omits landings and other design conditions. It is not a determination of compliance with Australian accessibility requirements, which depend on the ramp type and setting.
The ramp rises 1.2 m. Design two valid model options. For each:
(i) Choose a positive angle θ no greater than tan⁻¹(1/14).
(ii) Calculate the sloped length L = 1.2/sin θ (2 d.p.).
(iii) Confirm that L ≤ 18 m.
(iv) Recommend one option, using the gradient and length to explain the trade-off.
How did this worksheet feel?
What I'll revisit before next class:
1.1, hyp = 22, θ = 42°
opp = 22 sin 42° ≈ 22 × 0.6691 ≈ 14.72.
Pythagoras: adj² = 22² − 14.72² ≈ 484 − 216.68 ≈ 267.32, so adj ≈ 16.35.
1.2, opp = 8.4, θ = 65°
hyp = 8.4 / sin 65° ≈ 8.4 / 0.9063 ≈ 9.27.
1.3, Flagpole, sun at 32°, pole = 12 m
opp = 12, want hyp (string-to-shadow-tip). hyp = 12 / sin 32° ≈ 12 / 0.5299 ≈ 22.64 m.
1.4, Ramp X (18°) vs Ramp Y (36°), both 10 m
Ramp X rise = 10 sin 18° ≈ 3.09 m.
Ramp Y rise = 10 sin 36° ≈ 5.88 m.
Factor of increase ≈ 5.88 / 3.09 ≈ 1.90 NOT exactly 2. Sine is not linear: doubling the angle roughly doubles sin θ only for small angles. As angles grow, sin θ grows more slowly.
1.5, Slope: 50 m walked, 8 m up
sin θ = opp/hyp = 8/50 = 0.16. θ = sin⁻¹(0.16) ≈ 9° (to nearest degree).
1.6, Two-section slide
Section 1 length = 6 m.
Section 2 length = 2 / sin 60° ≈ 2 / 0.8660 ≈ 2.31 m.
Total ≈ 6 + 2.31 = 8.31 m.
2, Find the mistake
(a) The mistake is on Line 3.
(b) From sin 40° = 9 / hyp, the student multiplied both sides by sin 40° instead of by hyp. To isolate hyp, you must first multiply both sides by hyp (giving sin 40° × hyp = 9) and THEN divide by sin 40°. The correct rearrangement is hyp = 9 / sin 40°, not 9 × sin 40°.
(c) Corrected from Line 3:
hyp = 9 / sin 40°
≈ 9 / 0.6428
≈ 14.00 m.
Sense-check: hyp (14.00) > opp (9), which matches the rule "hyp is the longest side". The student's answer of 5.79 m was shorter than opp, a clear red flag.
3, Wheelchair-access ramp design (sample solutions)
The model limit is θ ≤ tan⁻¹(1/14) ≈ 4.0856°. The sloped length is L = 1.2 / sin θ; a smaller θ gives a longer section.
Option A: θ = 4°. Its gradient is tan 4° ≈ 0.0699, which is less than 1/14 ≈ 0.0714. L = 1.2 / sin 4° ≈ 17.20 m, within the model's 18 m sloped-length limit.
Option B: θ = 3.9°. Its gradient is tan 3.9° ≈ 0.0682, also less than 1/14. L = 1.2 / sin 3.9° ≈ 17.64 m, within the model's 18 m sloped-length limit.
Recommendation: Either option fits the stated maths model. Option B is gentler but uses about 0.44 m more sloped length than Option A.
Marking: 1 mark per valid option (angle satisfies θ ≤ tan⁻¹(1/14), L computed correctly, L ≤ 18 m verified); 1 mark for a recommendation supported by the gradient/length trade-off; 1 mark for clear setup using hyp = opp/sin θ. This simplified model does not establish accessibility compliance.