IGCSE Mathematics: Trigonometry — Revision Notes

Condensed for the final weeks. Pair these notes with the Trigonometry practice questions for worked exam-style application.

Pythagoras’ theorem

hyp^2 = a^2 + b^2

Applies only to right-angled triangles. The hypotenuse is always the longest side, always opposite the right angle – identify it first before deciding whether to add the two shorter sides’ squares (finding the hypotenuse) or subtract one square from the other (finding a shorter side).

Right-angled triangles: SOHCAHTOA

sin(angle) = opposite / hypotenuse
cos(angle) = adjacent / hypotenuse
tan(angle) = opposite / adjacent

Relative to the marked angle, not a fixed position on the page – the opposite and adjacent sides swap depending on which angle in the triangle is being used, so identify the marked angle first, then label the three sides relative to it.

Exact trigonometric values

angle	0	30	45	60	90
sin	0	1/2	sqrt(2)/2	sqrt(3)/2	1
cos	1	sqrt(3)/2	sqrt(2)/2	1/2	0
tan	0	sqrt(3)/3	1	sqrt(3)	undefined

An “exact value” answer must not be a rounded decimal approximation of an irrational value – writing 0.707 instead of √2/2 loses accuracy and is not acceptable. This is different from a terminating decimal that equals the exact value precisely: 0.5 is exactly equal to 1/2 (not a rounding of it), so it is an acceptable exact answer for sin 30 degrees unless the question specifically demands fraction notation.

Trigonometric functions and their graphs

The graphs of y = sin x and y = cos x are smooth waves repeating every 360 degrees, both oscillating between -1 and 1. y = cos x starts at its maximum (cos 0 = 1), falls to its minimum at 180 degrees (cos 180 = -1), and returns to its maximum at 360 degrees. y = sin x starts at 0, rises to its maximum at 90 degrees, and falls back through 0 at 180 degrees. y = tan x repeats every 180 degrees and is undefined at 90 and 270 degrees.

Non-right-angled triangles

Sine rule:    a / sin A = b / sin B = c / sin C
Cosine rule:  a^2 = b^2 + c^2 - 2bc cos A
Area:         Area = (1/2) ab sin C

Use the sine rule when a matching angle-side pair is known (an angle and the side opposite it). Use the cosine rule when either two sides and the included angle are known, or all three sides are known and an angle is required. The area formula needs two sides and the angle between them.

Sine rule example: angle A = 40, angle B = 65, side a = 8 cm. Find side b.
b / sin B = a / sin A
b = (a x sin B) / sin A = (8 x sin 65) / sin 40 = 11.3 cm (3 s.f.)

Cosine rule example: PQ = 7 cm, QR = 9 cm, angle Q = 55 degrees. Find PR.
PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)cos Q = 49 + 81 - 126 cos 55 = 57.7
PR = 7.60 cm (3 s.f.)

The cosine rule can also be rearranged to find an angle when all three sides are known: cos A = (b^2 + c^2 - a^2) / (2bc). A negative value inside cos-inverse is not an error – it correctly signals an obtuse angle.

Pythagoras and trigonometry in 3D

3D problems are solved by finding the correct 2D right-angled triangle inside the solid first – typically a diagonal across a face or the base – then applying Pythagoras or SOHCAHTOA within that flat triangle.

Cuboid 6 x 8 x 10:
base diagonal   = sqrt(6^2 + 8^2) = sqrt(100) = 10 cm
space diagonal  = sqrt(6^2 + 8^2 + 10^2) = sqrt(200) = 14.1 cm (3 s.f.)
angle to base: tan(angle) = height / base diagonal = 10 / 10 = 1 -> angle = 45.0 degrees

Keeping an intermediate length in exact (surd) form, rather than rounding it early, avoids losing accuracy before the final step.

Exam traps

• Applying Pythagoras to the wrong pair of sides, forgetting the hypotenuse is always opposite the right angle.
• Mixing up which side is opposite, adjacent and hypotenuse relative to the marked angle in an unfamiliar orientation.
• Giving an exact-value answer as a rounded decimal approximation of an irrational surd value (e.g. 1.73 instead of sqrt(3)) – a genuinely exact terminating decimal such as 0.5 for 1/2 is not this error.
• Using the sine rule when the cosine rule is needed (or the reverse), because the given information doesn’t match a sine-rule angle-side pair.
• In a 3D problem, applying a 2D triangle directly to the solid without first identifying the correct base diagonal or cross-section.
• Rounding an intermediate length before the final trigonometric step, which compounds into an inaccurate final answer.

Self-test

1. State which side of a right-angled triangle is always the hypotenuse.
2. Write down the exact value of tan 45 degrees.
3. State the condition needed to use the sine rule rather than the cosine rule.
4. Write down the formula for the area of a triangle given two sides and the included angle.
5. In a 3D cuboid problem, what should be found first before calculating the space diagonal?
6. Write down the rearranged cosine rule formula used to find an angle, given all three sides of a triangle.

Answers: 1. The longest side, always opposite the right angle. 2. tan 45 = 1. 3. A known angle and the side directly opposite it (a matching angle-side pair). 4. Area = (1/2) ab sin C, where C is the angle between sides a and b. 5. The base diagonal (the 2D diagonal across the cuboid’s base), found using Pythagoras’ theorem on the two base edges. 6. cos A = (b^2 + c^2 - a^2) / (2bc).