Mathematics • Year 9 • Unit 4 • Lesson 1
SOH CAH TOA — Mixed Challenge
Pull together everything from Lesson 1: labelling sides, picking the right ratio, evaluating exact and approximate trig values, and using inverse trig. You'll also spot a mistake in someone else's working and tackle an open-ended challenge.
1. Mixed problems — choose the right tool
Each question uses a different combination of the trig ideas from Lesson 1. Decide which ratio applies before you start writing. Show your working. Calculator in degree mode. 3 marks each
1.1 A right-angled triangle has angle 60° and opposite side 9 cm. Find the hypotenuse to 2 dp.
1.2 A right-angled triangle has opposite = 7 and adjacent = 24. Find the angle θ to 1 dp.
1.3 Find the EXACT values of sin 45°, cos 45° and tan 45°. (Leave surds in your answer.)
1.4 A right-angled triangle has hypotenuse 13 cm and one angle is 22.6°. Find both other sides to 1 dp.
1.5 If cos θ = 0.6 and θ is acute, find θ to the nearest degree. Then find sin θ using your triangle (no calculator for sin).
1.6 A right-angled triangle has one acute angle of 30°. The side adjacent to that angle is 5√3 cm. Find the opposite side and hypotenuse, leaving exact values (no decimals).
2. Find the mistake
Another student has tried to find the opposite side of a right-angled triangle with angle 40° and hypotenuse 15 cm. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find the opposite side when angle = 40° and hypotenuse = 15 cm:
Line 1: We have angle 40° and hypotenuse 15. We want opposite.
Line 2: SOH → sin = opposite / hypotenuse.
Line 3: sin 40° = opp / 15
Line 4: opp = 15 / sin 40° = 15 / 0.6428 ≈ 23.3 cm
Line 5: So opposite ≈ 23.3 cm.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: notice the opposite would be longer than the hypotenuse — which is impossible.)
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? To get from sin 40° = opp / 15 to a value for opp, do you DIVIDE 15 by sin 40°, or MULTIPLY 15 by sin 40°?3. Open-ended challenge — design your own triangle
This question has more than one valid answer — there are several triangles that work. 4 marks
3.1 Design two different right-angled triangles in which one of the trig ratios equals exactly 3/5 (i.e. sin θ = 3/5, OR cos θ = 3/5, OR tan θ = 3/5 — your choice for each triangle).
For each triangle you design:
(i) State which ratio equals 3/5 and which angle θ is involved.
(ii) Give the lengths of all three sides (whole numbers, all the same units).
(iii) Verify using Pythagoras (or sin² + cos² = 1) that your triangle is genuinely right-angled.
(iv) Find the angle θ to 1 dp.
Bonus: Your two triangles must use DIFFERENT ratios (e.g. one uses sin, the other tan).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Angle 60°, opposite 9, find hypotenuse
sin 60° = 9 / hyp → hyp = 9 / sin 60° = 9 / 0.8660 ≈ 10.39 cm.
Sanity check: hypotenuse is longer than the opposite (9). ✓
1.2 — Opposite 7, adjacent 24, find angle
tan θ = 7 / 24 = 0.2917 → θ = tan⁻¹(0.2917) ≈ 16.3°.
1.3 — Exact values at 45°
sin 45° = 1/√2 = √2/2. cos 45° = 1/√2 = √2/2. tan 45° = 1. (From a 1-1-√2 right-angled triangle.)
1.4 — Hypotenuse 13, angle 22.6°
Opposite = 13 × sin 22.6° = 13 × 0.3843 ≈ 5.0 cm.
Adjacent = 13 × cos 22.6° = 13 × 0.9232 ≈ 12.0 cm.
This is the famous 5-12-13 Pythagorean triple — check: 5² + 12² = 169 = 13². ✓
1.5 — cos θ = 0.6
θ = cos⁻¹(0.6) ≈ 53°.
Build a triangle: cos θ = adj/hyp = 0.6 = 3/5, so adj = 3, hyp = 5. By Pythagoras opp = √(25 − 9) = 4.
So sin θ = opp/hyp = 4/5 = 0.8. (No calculator needed for sin.)
1.6 — Angle 30°, adjacent 5√3
Opposite = adjacent × tan 30° = 5√3 × (1/√3) = 5 cm.
Hypotenuse: cos 30° = adj/hyp → hyp = adj / cos 30° = 5√3 / (√3/2) = 5√3 × (2/√3) = 10 cm.
This is the famous 30-60-90 triangle: sides in ratio 1 : √3 : 2. With adj = 5√3, opp = 5, hyp = 10.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) From sin 40° = opp / 15, you MULTIPLY both sides by 15 to make opp the subject — you don't divide 15 by sin 40°. The student's answer of 23.3 cm is longer than the hypotenuse (15 cm), which is impossible — that's a red flag the working is wrong.
(c) Corrected working:
sin 40° = opp / 15
opp = 15 × sin 40°
opp = 15 × 0.6428
opp ≈ 9.64 cm.
9.64 cm is less than the hypotenuse (15 cm) — that's the sanity check the student should have done.
3 — Open-ended challenge (sample solutions)
Many valid answers; here are two using DIFFERENT ratios:
Triangle A: sin θ = 3/5. Build a triangle with opposite = 3, hypotenuse = 5. By Pythagoras adjacent = √(25 − 9) = 4. So sides are 3, 4, 5. Verify: 3² + 4² = 25 = 5². ✓ Angle θ = sin⁻¹(3/5) = sin⁻¹(0.6) ≈ 36.9°.
Triangle B: tan θ = 3/5. Build a triangle with opposite = 3, adjacent = 5. By Pythagoras hypotenuse = √(9 + 25) = √34. So sides are 3, 5, √34. Verify: 3² + 5² = 34 = (√34)². ✓ Angle θ = tan⁻¹(3/5) = tan⁻¹(0.6) ≈ 31.0°.
Other valid approaches: Triangle C with cos θ = 3/5 → adj = 3, hyp = 5, opp = 4 (a relabelled 3-4-5 triangle) and θ ≈ 53.1°. Multiples like 6-8-10, 9-12-15 also give sin = 3/5.
Marking: 2 marks per triangle (1 for correct side lengths + Pythagoras check, 1 for the angle to 1 dp). Up to 4 in total. The triangles must use different ratios.