IB MYP Mathematics: Trigonometry and Pythagoras in MYP (MYP Mathematics) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

MYP mathematics is judged against criteria A to D, not by marks. The marks here are only a way to check your own working. The questions cover the trigonometry rows of spatial reasoning in the IB Mathematics guide (for use from September 2020/January 2021), first assessment 2022. Questions marked (Extended) match MYP 4-5 (extended). Questions marked (Enrichment) match the framework’s enrichment list. The rest are MYP 4-5 (standard).

Revise first with the study guide and revision notes. See the MYP mathematics hub, the checklist and the free diagnostics.

Questions

1. The angles of a triangle are (3x + 5)°, (3x + 5)° and (x + 30)°. Find x, the size of each angle, and say whether the triangle is acute, right-angled or obtuse. [3]

2. Two sides of a triangle are 8 cm and 5 cm. The third side, k cm, is a whole number. Find the smallest and largest possible values of k. [3]

3. Town B is 18 km from town A on a bearing of 125°. (a) Write the bearing of A from B. [1] (b) Find how far B is east of A and how far south of A, each to 3 s.f. [3]

4. A rectangular field is 96 m long and 40 m wide. Ana walks along the two sides from one corner to the opposite corner. Dev walks along the diagonal. How much shorter is Dev’s route? [4]

5. (Extended) (a) Show that a triangle with sides 2.5 m, 6 m and 6.5 m is right-angled, and say which side is opposite the right angle. [3] (b) Is a triangle with sides 4.2 m, 5.6 m and 7.1 m right-angled? Justify. [1]

6. A 6.5 m ladder leans against a vertical wall. Its foot is 2.2 m from the wall on level ground. (a) Find the angle the ladder makes with the ground, to 1 d.p. [2] (b) Find how high up the wall the ladder reaches, to 3 s.f. [2] (c) Give one check that shows your height is sensible. [1]

7. From the top of a cliff 45 m above the sea, the angle of depression of a boat is 28°. (a) Find the horizontal distance from the cliff to the boat, to 3 s.f. [2] (b) The boat moves 20 m towards the cliff. Find the new angle of depression, to 1 d.p. [3]

8. (Extended) In triangle ABC, a = 10.5 cm, b = 16 cm and angle A = 35°. (a) Find the two possible sizes of angle B, to 1 d.p. [3] (b) Explain why both are possible. [1] (c) Find side c for the triangle in which B is obtuse, to 3 s.f. [2]

9. (Extended) A triangle has sides 9 cm, 12 cm and 14 cm. Find its largest angle to 1 d.p. and say why you chose that angle. [4]

10. (Extended) Two boats leave port P together. Boat A sails 14 km on a bearing of 038°. Boat B sails 20 km on a bearing of 115°. (a) Find angle APB. [1] (b) Find the distance AB, to 3 s.f. [3] (c) Find the bearing of B from A, to the nearest degree. [4]

11. (Enrichment) (a) Find the distance between (2, −1, 4) and (6, 3, −3). [2] (b) Show that (−5, 12) lies on the circle x² + y² = 169. Then write the equation of the circle with centre at the origin that passes through (8, −6). [3] (c) Write 135° in radians as a multiple of π. [1] (d) Angle θ is obtuse and sin θ = 5/13. Use sin²θ + cos²θ = 1 to find cos θ and tan θ. [3]

Answers

1. 2(3x + 5) + (x + 30) = 180 [1], so 7x = 140 and x = 20 [1]. The angles are 65°, 65° and 50°, so the triangle is acute (and isosceles) [1]. [3] Examiner insight: Check that the angles you find add to 180°, and name the type with a reason, since “acute” is only justified once all three angles are known.

2. The third side lies between 8 − 5 and 8 + 5, so 3 < k < 13 [1]. Smallest whole number k = 4 [1]. Largest k = 12 [1]. [3] Examiner insight: Be careful with strict inequalities: k = 3 gives a flat triangle and k = 13 as well, so neither counts.

3. (a) 125° + 180° = 305° [1] (b) The angle from the south line is 180° − 125° = 55°. East = 18 sin 55° [1] = 14.7 km [1]. South = 18 cos 55° = 10.3 km [1]. [4] Examiner insight: A sketch with the north line and the 55° angle marked shows which ratio you chose and why.

4. Diagonal² = 96² + 40² [1], so diagonal = 104 m [1]. Ana walks 96 + 40 = 136 m [1]. Dev saves 136 − 104 = 32 m [1]. [4] Examiner insight: Show the comparison in a final sentence with units, since the question asks for a difference, not a distance.

5. (a) 2.5² + 6² = 6.25 + 36 = 42.25 [1]. 6.5² = 42.25 [1]. They are equal, so the triangle is right-angled, with the right angle opposite the 6.5 m side [1]. (b) 4.2² + 5.6² = 49.00 but 7.1² = 50.41, so they differ and it is not right-angled [1]. [4] Examiner insight: Evaluate both sides of the equation and compare them. A bare “yes” or “no” does not show the test.

6. (a) cos θ = 2.2/6.5 [1], so θ = 70.2° [1]. (b) Height² = 6.5² − 2.2² = 37.41 [1], so height = 6.12 m [1]. (c) The height 6.12 m is less than the ladder length 6.5 m, as it must be, since the ladder is the hypotenuse [1]. [5] Examiner insight: A reasonableness check is a short sentence linking the number to the situation, which is what criterion D asks for.

7. (a) tan 28° = 45/d [1], so d = 45 ÷ tan 28° = 84.6 m [1]. (b) New distance = 84.63 − 20 = 64.63 m [1]. tan θ = 45/64.63 [1], so θ = 34.8° [1]. [5] Examiner insight: Keep the unrounded 84.63 in the second part. Using the rounded 84.6 can shift the final angle.

8. (a) sin B = 16 sin 35° ÷ 10.5 [1] = 0.874 [1]. B = 60.9° or 180° − 60.9° = 119.1° [1]. (b) 35° + 119.1° = 154.1°, which is less than 180°, so both angles fit in a triangle [1]. (c) C = 180° − 35° − 119.1° = 25.9° [1]. c = 10.5 × sin 25.9° ÷ sin 35° = 8.00 cm [1]. [6] Examiner insight: After taking the inverse sine, always ask whether 180° minus that angle also works. Missing it loses the second solution.

9. The largest angle is opposite the longest side (14 cm) [1]. cos θ = (9² + 12² − 14²) ÷ (2 × 9 × 12) [1] = 29/216 [1]. θ = 82.3° [1]. [4] Examiner insight: Choosing the angle opposite the longest side, and saying so, shows why this angle is the largest. A positive cosine confirms it is acute.

10. (a) 115° − 38° = 77° [1] (b) AB² = 14² + 20² − 2(14)(20)cos 77° [1] = 470.0 [1], so AB = 21.7 km [1]. (c) sin PAB ÷ 20 = sin 77° ÷ 21.68 [1], so angle PAB = 64.0° [1]. The bearing of P from A is 38° + 180° = 218° [1]. From a sketch, B lies 64.0° anticlockwise from AP, so the bearing is 218° − 64.0° = 154° [1]. [8] Examiner insight: Bearings are often lost at the last step. A sketch with a north line at A shows the direction in which to add or subtract the angle.

11. (a) √[(6 − 2)² + (3 + 1)² + (−3 − 4)²] = √(16 + 16 + 49) [1] = 9 [1]. (b) (−5)² + 12² = 25 + 144 = 169, so the point lies on the circle [1]. For (8, −6), r² = 64 + 36 = 100 [1], so the equation is x² + y² = 100 [1]. (c) 135° = 135π/180 = 3π/4 [1]. (d) cos²θ = 1 − (5/13)² = 144/169 [1]. θ is obtuse, so cos θ is negative: cos θ = −12/13 [1]. tan θ = sin θ ÷ cos θ = −5/12 [1]. [9] Examiner insight: The sign is the usual slip in part (d). Say that an obtuse angle has a negative cosine before you choose the root.

Where marks are usually lost

• Using Pythagoras by adding when a shorter side is missing.
• Leaving the angle mode on radians when the question uses degrees.
• Skipping the sketch on bearings questions, then measuring from the wrong direction.
• Rounding early and carrying the rounded value into the next step.
• Forgetting the second answer from the sine rule.
• Treating the converse test as a statement rather than a comparison of two values.
• Choosing the sine rule when the known angle is not opposite a known side.
• Giving a final answer without units or without the degree sign.
• Not checking that an answer fits the context, which weakens criterion D.
• Leaving out the sign of cosine and tangent for obtuse angles.

Next steps

• Revise the method: revision notes
• Learn the content in full: study guide
• Course page: MYP mathematics hub
• Tick off topics: printable checklist
• Find weak areas: all free 10-minute diagnostics
• Book a free trial class: trial

Official syllabus

IB Mathematics guide (for use from September 2020/January 2021), first assessment 2022, International Baccalaureate Organization: spatial reasoning framework, trigonometry rows.