OCR GCSE Mathematics: Pythagoras, right-angled trigonometry and exact trigonometric values (J560) – Revision Notes

These notes condense OCR GCSE (9-1) Mathematics J560, specification Version 2.1 (August 2026), statements 10.05a, 10.05b and 10.05c. The study guide has the full explanations and longer worked examples. Test yourself with the practice questions, see the OCR GCSE Mathematics hub and the printable checklist, and take a free 10-minute diagnostic to find gaps. The 2D work and exact values are in the Foundation tier and Higher tier content. Complex figures and 3D figures are Higher tier only.

Key facts

Ref	Fact	Tier
10.05a	a² + b² = c², where c is the hypotenuse	Foundation and Higher
10.05a	Use it in complex figures and 3D figures	Higher tier only
10.05b	sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj	Foundation and Higher
10.05b	Use them in complex figures and 3D figures	Higher tier only
10.05c	Exact values for 0°, 30°, 45°, 60°, 90° (tan: not 90°)	Foundation and Higher

Pythagoras: method in steps

1. Mark the right angle. The side opposite it is the hypotenuse, c.
2. Hypotenuse unknown: add the squares, then square root.
3. Short side unknown: subtract the square of the known short side from the square of the hypotenuse, then square root.
4. Check that the hypotenuse is the longest side in your answer.

Short sides 5 and 12:   c² = 25 + 144 = 169,  c = 13
Hypotenuse 10, side 6:  a² = 100 − 36 = 64,   a = 8

Converse test. If a² + b² = c² for the three sides, the triangle is right-angled.

Where it hides. Isosceles triangle (split down the middle), rectangle diagonal, rhombus diagonals, distance between two coordinates (horizontal step and vertical step), ladder against a wall.

Derivation reminder. Four identical right-angled triangles in a square of side (a + b) leave a tilted square of side c: (a + b)² = 4(½ab) + c², which simplifies to a² + b² = c².

Trigonometry: method in steps

1. Label the sides hypotenuse, opposite, adjacent from the angle you are using.
2. Write the ratio that contains the known side and the wanted side.
3. Unknown on top: multiply. Unknown on the bottom: divide.
4. For an angle, use sin⁻¹, cos⁻¹ or tan⁻¹ on the ratio.

Find opposite:        x = 20 × sin 35°
Find hypotenuse:      h = 6.4 / cos 48°
Find angle:           θ = tan⁻¹(7/9)

Know	Want	Use
Hyp	Opp	sin, multiply
Hyp	Adj	cos, multiply
Adj	Opp	tan, multiply
Opp	Hyp	sin, divide
Adj	Hyp	cos, divide
Opp	Adj	tan, divide

Similar triangles. Right-angled triangles with the same angles have the same ratios, which is why sin, cos and tan depend on the angle alone (see 9.04c).

Elevation and depression. Both angles start at the horizontal line of sight. Elevation turns upward and depression turns downward.

Exact values

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	√3/3	1	√3	not required

Rebuild it in 10 seconds.

• Right-angled isosceles triangle, short sides 1, hypotenuse √2: gives the 45° row.
• Equilateral triangle of side 2 cut in half: sides 1, √3, 2 with angles 30° and 60°: gives the other rows.
• sin 0°, sin 30°, sin 45°, sin 60°, sin 90° read 0, 1/2, √2/2, √3/2, 1 (that is √0/2 up to √4/2). Cos runs backwards through the same list.

Exact answers. Leave surds in the answer, and rationalise a denominator if one remains: 1/√2 = √2/2.

Triangle with a 30° angle, hypotenuse 10:
opposite = 10 × sin 30° = 5
adjacent = 10 × cos 30° = 5√3

Higher tier only: complex figures and 3D

1. Sketch the 3D shape and mark the right angles.
2. Pull out one flat right-angled triangle at a time and redraw it with the lengths on.
3. Find the missing length, then use it in the next triangle. Keep exact values (such as √125) for as long as possible.
4. For the steepness of a line against a plane, use the line, its shadow on the plane, and the vertical drop.

Cuboid 8 × 6 × 5:
base diagonal = √(64 + 36) = 10
space diagonal = √(100 + 25) = √125 = 11.2
angle to base = tan⁻¹(5/10) = 26.6°

For a pyramid with its apex above the centre of a square base, the horizontal distance to a corner is half the base diagonal, and to the middle of a base edge is half the base side.

Worked reminders

Rectangle diagonal. A rectangle is 7.5 cm by 4 cm. Its diagonal is the hypotenuse of a right-angled triangle.

d² = 7.5² + 4² = 56.25 + 16 = 72.25
d = 8.5 cm

Angle from two sides. Hypotenuse 11 cm, opposite 6.3 cm. The two sides involved are opposite and hypotenuse, so use sine.

sin θ = 6.3/11
θ = sin⁻¹(6.3/11) = 34.9° (1 d.p.)

Angle of elevation. You stand 40 m from the foot of a building. The angle of elevation of the top is 28°. The horizontal distance is the adjacent side and the height is the opposite side.

height = 40 × tan 28° = 21.3 m (3 s.f.)

Exact working. Show that sin 60° ÷ tan 60° = 1/2.

sin 60° ÷ tan 60° = (√3/2) ÷ √3 = 1/2

Must-know distinctions

Idea	Do not confuse with
Hypotenuse: opposite the right angle, always the longest side	Opposite: opposite the working angle, which changes when you switch angle
Short side unknown: subtract squares	Hypotenuse unknown: add squares
Unknown on top of the ratio: multiply	Unknown on the bottom: divide
tan θ = 9/14 gives the ratio	tan⁻¹(9/14) gives the angle
sin 30° = 1/2	sin 60° = √3/2
cos 30° = √3/2	cos 60° = 1/2
Angle of elevation: up from the horizontal	Angle of depression: down from the horizontal
Exact value: a surd or fraction	Decimal value: rounded, so it does not answer a question that asks for an exact value

Units, rounding and checks

• Give lengths with units (cm, m) and areas with square units. Angles take the degree sign.
• Round lengths to 3 significant figures and angles to 1 decimal place unless the question says otherwise.
• Sense-check every answer. The hypotenuse must be longer than either short side, and an angle in a right-angled triangle must be less than 90°.
• If a trig value looks odd, check that you are working in degrees and not radians.
• Where a question asks you to show a result, write every line of working down to the stated answer.

Quick self-test

1. Find the hypotenuse of a right-angled triangle with short sides 9 and 40.
2. A right-angled triangle has hypotenuse 26 and one short side 10. Find the other short side.
3. Find x if sin 30° = x/18.
4. Find the length of the adjacent side if the hypotenuse is 20 and the angle is 35° (3 s.f.).
5. Find the angle θ if tan θ = 1.2 (1 d.p.).
6. A right-angled triangle has an angle of 40° and an opposite side of 7 cm. Find the adjacent side (3 s.f.).
7. Write down the exact value of tan 30° ÷ sin 30°.
8. Write down the exact value of sin 45° × cos 45°.
9. A right-angled triangle has hypotenuse 9 cm and opposite side 4.5 cm. Find the angle opposite the 4.5 cm side without a calculator.
10. (Higher tier only) A cuboid is 2 cm by 3 cm by 6 cm. Find the length of the space diagonal.

Answers

1. c² = 81 + 1600 = 1681, so 41.
2. a² = 676 − 100 = 576, so 24.
3. x = 18 × 1/2 = 9.
4. 20 × cos 35° = 16.4.
5. tan⁻¹(1.2) = 50.2°.
6. 7 / tan 40° = 8.34 cm.
7. (√3/3) ÷ (1/2) = 2√3/3.
8. (√2/2) × (√2/2) = 2/4 = 1/2.
9. sin θ = 4.5/9 = 1/2, so θ = 30°.
10. √(4 + 9 + 36) = √49 = 7 cm.

Where marks are usually lost

• Writing a bare final answer on a multi-step problem. Show each triangle and each substitution so that a slip costs you less.
• Using the square root of one side minus another without squaring first: a² − b² is not (a − b)².
• Forgetting the last square root and giving c² as the length.
• Picking the wrong ratio because the sides were labelled from the right angle instead of from the working angle.
• Multiplying by sin when the unknown is the hypotenuse. Check whether the unknown is top or bottom.
• Rounding inside a long calculation, then giving an answer that is out in the third significant figure.
• Answering an exact-value question with a decimal. When the wording says exact, leave a surd or fraction.
• Mixing up the two shapes behind the exact values: the 1, 1, √2 triangle is 45°, while the 1, √3, 2 triangle is 30° and 60°.
• In 3D, using the wrong horizontal distance. A sloping edge of a pyramid goes to a corner, not to the middle of a base edge.

Official syllabus

OCR GCSE (9-1) Mathematics J560, specification Version 2.1 (August 2026), first assessment in 2017, published by OCR. Statements 10.05a, 10.05b and 10.05c.