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.