Pythagoras' Theorem & Trigonometry

Pythagoras' Theorem & Trigonometry

Pythagoras' Theorem

In a right-angled triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides.

a² + b² = c² (where c is the hypotenuse)

Worked Example: Find the hypotenuse of a right-angled triangle with sides 5 cm and 12 cm.

• c² = 5² + 12² = 25 + 144 = 169
• c = √169 = 13 cm

Worked Example: The hypotenuse is 10 cm and one side is 6 cm. Find the other side.

• a² = 10² − 6² = 100 − 36 = 64
• a = √64 = 8 cm

Pythagoras in 3D (Higher): Find the space diagonal of a cuboid by applying the theorem twice.

Worked Example: Find the length AG in a cuboid 3 cm × 4 cm × 12 cm.

• First, find the base diagonal: d² = 3² + 4² = 25, d = 5
• Then: AG² = 5² + 12² = 25 + 144 = 169
• AG = 13 cm

Or directly: AG = √(3² + 4² + 12²) = √169 = 13 cm.

Trigonometry (SOH CAH TOA)

For a right-angled triangle, label the sides relative to a chosen angle θ:

• Opposite (O): the side opposite θ
• Adjacent (A): the side next to θ (not the hypotenuse)
• Hypotenuse (H): the longest side, opposite the right angle

The three trigonometric ratios:

• sin θ = Opposite / Hypotenuse (SOH)
• cos θ = Adjacent / Hypotenuse (CAH)
• tan θ = Opposite / Adjacent (TOA)

Finding a Side

Worked Example: Find the side marked x. Angle = 35°, hypotenuse = 14 cm, x is the opposite side.

• sin 35° = x/14
• x = 14 × sin 35° = 14 × 0.5736 = 8.03 cm (to 3 s.f.)

Worked Example: Angle = 50°, adjacent = 8 cm, find the hypotenuse.

• cos 50° = 8/h
• h = 8 / cos 50° = 8 / 0.6428 = 12.4 cm (to 3 s.f.)

Finding an Angle

Use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹.

Worked Example: Opposite = 7, adjacent = 10. Find the angle.

• tan θ = 7/10 = 0.7
• θ = tan⁻¹(0.7) = 35.0° (to 1 d.p.)

Exact Trigonometric Values

You must memorise these (no calculator):

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Sine Rule and Cosine Rule (Higher)

For any triangle (not just right-angled), label angles A, B, C and opposite sides a, b, c.

Sine rule: a/sin A = b/sin B = c/sin C

Use when you know an angle and its opposite side, plus one other angle or side.

Cosine rule: a² = b² + c² − 2bc cos A

Use when you know two sides and the included angle (finding a side) or all three sides (finding an angle). Rearranged: cos A = (b² + c² − a²) / 2bc.

Worked Example (Sine rule): In triangle ABC, angle A = 40°, angle B = 75°, side a = 8 cm. Find side b.

• 8/sin 40° = b/sin 75°
• b = 8 × sin 75° / sin 40° = 8 × 0.9659 / 0.6428 = 12.0 cm

Worked Example (Cosine rule): Sides b = 7, c = 9, angle A = 52°. Find side a.

• a² = 49 + 81 − 2(7)(9)cos 52° = 130 − 126 × 0.6157 = 130 − 77.6 = 52.4
• a = 7.24 cm

Area of a Triangle

Area = ½ × a × b × sin C (using two sides and the included angle)

Exam Tips

• Choose SOH, CAH or TOA based on which sides you have and need — label O, A, H first
• Make sure your calculator is in DEGREE mode
• For 3D Pythagoras, draw the right-angled triangle you are using — examiners want to see it
• The sine rule has an ambiguous case (two possible triangles) — be aware but this is rarely examined at GCSE