IB MYP Mathematics: Trigonometry and Pythagoras in MYP (MYP Mathematics)

This guide covers the trigonometry rows of the spatial reasoning branch in the IB Mathematics guide (for use from September 2020/January 2021), first assessment 2022. The framework places triangle properties, bearings, Pythagoras’ theorem and right-angled trigonometry in MYP 4-5 (standard). The converse of Pythagoras and the sine and cosine rules are listed for MYP 4-5 (extended). The framework’s enrichment list adds further ideas, covered at the end and labelled enrichment. MYP has no tiers.

Use it with the revision notes and the practice questions. Course pages: the MYP mathematics hub and printable checklist. For how work is judged, see criteria in practice. To find weak spots first, try the free diagnostics.

What this unit covers

Framework row	You should be able to	Label
Triangle properties	Use angle sums, exterior angles, isosceles and equilateral facts, side rules	Standard
Bearings	Give and use three-figure bearings and back bearings	Standard
Pythagoras’ theorem	Find a missing side; solve in context	Standard
Trigonometric ratios	Use sine, cosine, tangent for sides and angles	Standard
Converse of Pythagoras	Test whether a triangle is right-angled	Extended
Sine and cosine rules, with applications	Solve any triangle; use in bearings problems	Extended
3D coordinates, area rule, unit circle, radians, circle at the origin, identities	See the last section	Enrichment

The framework also names algorithmic thinking in trigonometry. In practice this means following a clear step-by-step method, and choosing the right tool from a decision list (see “A method for choosing”).

Triangle properties

• The interior angles add to 180°.
• An exterior angle equals the sum of the two interior angles opposite it.
• An isosceles triangle has two equal sides and the angles opposite them are equal. An equilateral triangle has three equal sides and three 60° angles.
• The longest side is opposite the largest angle.
• Triangle inequality: any two sides add to more than the third side.

Example 1. The angles of a triangle are (x + 20)°, 2x° and (3x − 20)°. Find them. (x + 20) + 2x + (3x − 20) = 180, so 6x = 180 and x = 30. The angles are 50°, 60°, 70°. Check: they add to 180°.

Example 2. An exterior angle is 125° and one opposite interior angle is 48°. The other opposite angle is 125 − 48 = 77°.

Example 3. Can 5 cm, 7 cm and 13 cm form a triangle? 5 + 7 = 12, which is less than 13, so no. Can 6, 7 and 12? 6 + 7 = 13 > 12, and the other pairs are larger, so yes.

Bearings

A bearing is an angle measured clockwise from north, written with three figures (so 045°, not 45°). To find a back bearing, add or subtract 180°.

Example 4. A boat leaves harbour H and sails 12 km on bearing 060° to point K. It then sails 9 km on bearing 150° to point P.

• At K, the bearing 150° is 90° clockwise from 060°, so the path turns through a right angle. HKP is a right-angled triangle.
• HP² = 12² + 9² = 225, so HP = 15 km.
• The angle at H between HK and HP is tan⁻¹(9/12) = 36.9°. The bearing of P from H is 060° + 36.9° = 096.9°.
• The bearing of H from P is 096.9° + 180° = 276.9°.

Always sketch a north line at each point. Clockwise angles are measured from that line.

Pythagoras’ theorem

In a right-angled triangle with hypotenuse c (the side opposite the right angle): a² + b² = c².

Example 5. A support wire of length 8.5 m is fixed to a pole at a point, and its foot is 4.0 m from the base of the pole. How high is the fixing point? Height² = 8.5² − 4.0² = 72.25 − 16 = 56.25, so height = 7.5 m. When finding a shorter side, you subtract.

Example 6. A rectangular screen is 40 cm wide and 25 cm high. Diagonal² = 40² + 25² = 2225, so the diagonal is 47.2 cm (3 s.f.).

Trigonometric ratios in right-angled triangles

Label the sides from the angle θ you are using: hypotenuse, opposite (across from θ) and adjacent (next to θ).

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

To find an angle, use the inverse: θ = sin⁻¹(…), cos⁻¹(…) or tan⁻¹(…). Check that your calculator is in degree mode.

Example 7. A ramp rises 1.2 m and makes 15° with the ground. Its length L is the hypotenuse and 1.2 is opposite, so sin 15° = 1.2/L, giving L = 1.2 ÷ sin 15° = 4.64 m.

Example 8. You stand 30 m from a tower. The angle of elevation of its top is 38° and your eye is 1.6 m above ground. Height above eye level = 30 × tan 38° = 23.44 m. Tower height = 23.44 + 1.6 = 25.0 m (3 s.f.).

Example 9. A slope rises 7 m over a horizontal distance of 24 m. tan θ = 7/24, so θ = tan⁻¹(7/24) = 16.3°.

An angle of depression is measured downward from the horizontal. It equals the angle of elevation from the lower point to the upper, because the two are alternate angles.

Converse of Pythagoras (extended)

If the three sides satisfy a² + b² = c² for the longest side c, the triangle is right-angled. If they do not, it is not.

Example 10. Is a triangle with sides 9, 40 and 41 right-angled? 9² + 40² = 81 + 1600 = 1681 and 41² = 1681. Yes, and the right angle is opposite the side of 41. For 7, 8 and 11: 49 + 64 = 113, but 11² = 121. Not right-angled.

Sine rule and cosine rule (extended)

For a triangle with sides a, b, c opposite angles A, B, C:

• Sine rule: a / sin A = b / sin B = c / sin C
• Cosine rule: a² = b² + c² − 2bc cos A, and cos A = (b² + c² − a²) / 2bc

Example 11 (sine rule, side). A = 48°, B = 71°, a = 14. Then b = 14 × sin 71° ÷ sin 48° = 17.8.

Example 12 (sine rule, ambiguous case). a = 9, b = 12, A = 40°. sin B = 12 × sin 40° ÷ 9 = 0.857. So B = 59.0° or B = 180° − 59.0° = 121.0°. Both are possible because 40° + 121.0° is still less than 180°. Then C = 81.0° or 19.0°, and c = 13.8 or 4.56. Only a given angle opposite the shorter given side can produce two triangles.

Example 13 (cosine rule, side). Two sides 7.2 and 9.5 enclose 58°. Third side² = 7.2² + 9.5² − 2(7.2)(9.5)cos 58° = 69.6, so the side is 8.34.

Example 14 (cosine rule, angle). Sides 6, 8, 11. cos C = (36 + 64 − 121)/(2 × 6 × 8) = −0.21875, so C = 102.6°. A negative cosine means an obtuse angle.

Link to trigonometric functions. For an obtuse angle, sin θ = sin(180° − θ), which is why the sine rule can give two answers. The cosine of an obtuse angle is negative.

Example 15 (application). Surveyors measure from A: B is 420 m away on bearing 072°, and C is 310 m away on bearing 135°. Angle BAC = 135° − 72° = 63°. BC² = 420² + 310² − 2(420)(310)cos 63° gives BC = 393 m.

Enrichment ideas

These appear in the framework’s enrichment list only.

• 3D coordinate geometry: the distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) is √[(Δx)² + (Δy)² + (Δz)²]. From (1, 2, 3) to (5, −2, 6): √(16 + 16 + 9) = √41 = 6.40.
• Area of a triangle: area = ½ab sin C. With sides 8 and 11 enclosing 35°: ½ × 8 × 11 × sin 35° = 25.2.
• Unit circle: a point on a circle of radius 1 at angle θ from the positive x-axis is (cos θ, sin θ). For 150°, sin = 0.5 and cos = −0.866.
• Radians: 180° = π radians. So 150° = 5π/6, and 2.4 rad = 137.5°.
• Circle with centre at the origin: x² + y² = r². The point (3, −4) lies on x² + y² = 25.
• Identities: sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. If sin θ = 3/5 and θ is acute, cos θ = 4/5 and tan θ = 3/4. If θ is obtuse, cos θ = −4/5.

A method for choosing

1. Sketch the situation and label what you know. Add north lines for bearings.
2. Is there a right angle? If yes, use Pythagoras for two sides and one unknown side, or sin, cos, tan when an angle is involved.
3. No right angle? Check what you know. A matching side and opposite angle pair, plus one more piece: use the sine rule. Two sides and the angle between them, or three sides: use the cosine rule.
4. Substitute, solve, then check the answer is sensible (longest side opposite largest angle).
5. Round at the end and give units or degrees.

Linking to the criteria

Criterion A rewards choosing and applying the right method. Criterion C rewards clear diagrams and reasoning. Criterion D asks you to justify the accuracy of a solution and whether it makes sense in context, so state your rounding and compare answers with the situation. A criterion B investigation might explore Pythagorean triples and generalise a rule.

Common errors

• Using a² + b² = c² when the unknown is a shorter side, instead of subtracting.
• Calculator in radian mode.
• Measuring bearings from the wrong north line or anticlockwise.
• Using the cosine rule’s minus sign wrongly, for instance computing (b² + c² − 2bc) first.
• Forgetting the second possible angle in the ambiguous case.
• Rounding early.

Official syllabus

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