Mathematics • Year 9 • Unit 3 • Lesson 10
Mixed Trigonometry — Mixed Challenge
Tie together everything from L6–L10: pick the right ratio for any side-pair, combine ratios with Pythagoras, work with exact values for the special angles 30°, 45° and 60°, spot a ratio-choice mistake and design a "find every side" problem of your own.
1. Mixed problems — every problem could be any ratio
Identify the side-pair, name the ratio, rearrange, compute. Round to 2 d.p. unless told. 3 marks each
1.1 A right triangle has θ = 50°, hyp = 9. Find opp.
1.2 A right triangle has θ = 25°, adj = 14. Find opp.
1.3 A right triangle has θ = 70°, opp = 18. Find hyp. Then find adj using Pythagoras.
1.4 Without a calculator, find opp for a right triangle with θ = 30° and hyp = 12. (Use sin 30° = 0.5 exactly.) Then find adj using cos 30° ≈ 0.8660.
1.5 A right triangle has θ = 45°, adj = 7. Find opp (use tan 45° = 1 exactly) and hyp (use cos 45° = √2/2 ≈ 0.7071).
1.6 Two right triangles share the angle θ = 60°. Triangle A has opp = 4 cm. Triangle B has hyp = 14 cm. Find the missing measurements: hyp in A, and both legs in B. (Use sin 60° ≈ 0.8660, cos 60° = 0.5, tan 60° ≈ 1.7321.)
2. Find the mistake
A student tries to find the adjacent side of a right triangle given the angle and the hypotenuse. Exactly one line contains a Year 9 ratio-choice mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A right triangle has θ = 35° and hyp = 20. Find adj.
Student's working:
Line 1: Known: hyp = 20, θ = 35°. Wanted: adj.
Line 2: Sides involved: adj and hyp.
Line 3: Ratio: tan (since I want to find adj).
Line 4: tan 35° = adj / 20
Line 5: adj = 20 × tan 35° ≈ 20 × 0.7002 ≈ 14.00
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. Focus on the SIDE-PAIR → RATIO rule from L10.
(c) Write out the corrected working from that line onward, including the corrected final answer.
Stuck? The pair adj + hyp matches COS, not tan. Tan is the leg-only ratio (opp + adj, no hyp). The student picked tan because it was the ratio they used most recently, ignoring the side-pair.3. Open-ended challenge — design a "find every side" problem
This question has many valid answers. 4 marks
3.1 Design a complete worked example that uses BOTH sin and cos in a single real-world scenario. Your design must include:
(i) A real-world scenario (e.g. ladder, ramp, kite, tree, slide) in one sentence.
(ii) A right triangle with a clearly stated angle θ (choose 25° < θ < 75°) and a clearly stated hypotenuse length (choose 5 m ≤ hyp ≤ 30 m).
(iii) The side-pair, ratio name and worked computation for the opposite side (use sin).
(iv) The side-pair, ratio name and worked computation for the adjacent side (use cos).
(v) A Pythagoras check: opp² + adj² ≈ hyp² (state the actual numbers and confirm).
(vi) A "sanity sentence" — a one-line check that your answer makes physical sense (e.g. "the ladder reaches about 4 m up the wall — a sensible height for a 5 m ladder leaning at 60°").
How did this worksheet feel?
What I'll revisit before next class:
1.1 — θ = 50°, hyp = 9
opp + hyp → sin. opp = 9 sin 50° ≈ 9 × 0.7660 ≈ 6.89.
1.2 — θ = 25°, adj = 14
opp + adj → tan. opp = 14 tan 25° ≈ 14 × 0.4663 ≈ 6.53.
1.3 — θ = 70°, opp = 18
opp + hyp → sin. hyp = 18 / sin 70° ≈ 18 / 0.9397 ≈ 19.16.
Pythagoras: adj² ≈ 19.16² − 18² ≈ 367.10 − 324 ≈ 43.10, so adj ≈ 6.57.
1.4 — θ = 30°, hyp = 12 (no calculator for sin)
opp = 12 sin 30° = 12 × 0.5 = 6 exactly.
adj = 12 cos 30° ≈ 12 × 0.8660 ≈ 10.39.
1.5 — θ = 45°, adj = 7
opp = 7 tan 45° = 7 × 1 = 7 exactly (45° → opp = adj).
hyp = 7 / cos 45° ≈ 7 / 0.7071 ≈ 9.90.
Exactly: hyp = 7√2 ≈ 9.899.
1.6 — Two triangles, θ = 60°
Triangle A (opp = 4): hyp = 4 / sin 60° ≈ 4 / 0.8660 ≈ 4.62 cm. adj = 4 / tan 60° ≈ 4 / 1.7321 ≈ 2.31 cm.
Triangle B (hyp = 14): opp = 14 sin 60° ≈ 14 × 0.8660 ≈ 12.12 cm. adj = 14 cos 60° = 14 × 0.5 = 7 cm exactly.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The side-pair was correctly identified on Line 2 as adj + hyp. The L10 decision rule says adj + hyp → cos, not tan. Tangent is the leg-only ratio (opp + adj, with NO hyp). The student appears to have picked tan by association with "want to find adj" instead of by the side-pair rule.
(c) Corrected from Line 3:
Ratio: cos (adj + hyp → cos, CAH).
cos 35° = adj / 20
adj = 20 × cos 35°
≈ 20 × 0.8192
≈ 16.38.
Sense-check: for θ < 45°, adj > opp (and adj close to hyp). 16.38 is close to 20 ✓. The student's wrong answer of 14.00 is the opp, not the adj.
3 — Design a "find every side" problem (sample solution)
(i) Scenario: A 12 m fire-truck ladder is set at 60° to the ground against a building wall.
(ii) Triangle: θ = 60°, hyp = 12 m, opp = height up the wall, adj = foot distance from the wall.
(iii) Opposite (height up wall): side-pair opp + hyp → sin. opp = 12 sin 60° ≈ 12 × 0.8660 ≈ 10.39 m.
(iv) Adjacent (foot from wall): side-pair adj + hyp → cos. adj = 12 cos 60° = 12 × 0.5 = 6.00 m.
(v) Pythagoras check: opp² + adj² = 10.39² + 6.00² ≈ 107.95 + 36.00 ≈ 143.95 ≈ 12² = 144 ✓ (tiny rounding error).
(vi) Sanity sentence: A 12 m ladder at 60° reaches about 10.4 m up the wall with its foot 6 m out — a steep, tall reach. That matches what a fire crew uses to reach a 3rd-storey window.
Marking: 1 mark for parts (iii) and (iv) computed correctly with side-pair + ratio name; 1 mark for the Pythagoras check; 1 mark for a sensible scenario and sanity sentence; 1 mark for full presentation.