IGCSE Mathematics: Trigonometry (Cambridge 0580)

This guide covers Topic 6 Trigonometry, for Cambridge IGCSE Mathematics 0580, 2025–2027 series. The Core subtopics (C6.1–C6.2) are examined at all entry levels; the Extended-only subtopics (E6.2 additions, plus E6.3, E6.4, E6.5 and E6.6, none of which have a Core equivalent) are required only for the Extended tier. Extended candidates are eligible for grades A* to E and Core candidates for grades C to G, so grades A* to B need the Extended tier (the syllabus advises Extended for anyone aiming at grades A* to C).

Where this fits in 0580

Trigonometry is the sixth of nine topics in 0580, and it builds directly on two earlier topics: Number’s work with ratio and decimals, and Geometry’s angle facts and constructions. Coordinate geometry’s gradient work (Topic 3) also resurfaces here, since a right-angled triangle drawn between two coordinates is the usual route to finding a line’s length. Bearings link it to Geometry: three-figure bearings are taught in C4.3/E4.3 Scale drawings, and the notes to C6.2/E6.2 say that knowledge of bearings may be required when solving two-dimensional problems with Pythagoras’ theorem and trigonometry. At Extended tier, E6.6 extends these methods to three dimensions, including the angle between a line and a plane, so familiarity with 3D solids from Mensuration helps.

Syllabus coverage

CAMBRIDGE IGCSE MATHEMATICS 0580 — TOPIC 6 TRIGONOMETRY

• C6.1 Pythagoras’ theorem — know and use Pythagoras’ theorem
• C6.2 Right-angled triangles — know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a right-angled triangle; solve problems in two dimensions using Pythagoras’ theorem and trigonometry; angles are given in degrees, with answers written in degrees to one decimal place; knowledge of bearings may be required

Note that C6.3, C6.4, C6.5 and C6.6 exist only as Extended-tier subtopics (E6.3–E6.6 below) — there is no Core content at those numbers.

Extended only (in addition to the Core content above)

• E6.2 Right-angled triangles (Extended) — as C6.2, plus knowing that the perpendicular distance from a point to a line is the shortest distance to the line, and carrying out calculations involving angles of elevation and depression
• E6.3 Exact trigonometric values — knowing the exact values of sin x and cos x for x = 0°, 30°, 45°, 60° and 90°, and of tan x for x = 0°, 30°, 45° and 60°
• E6.4 Trigonometric functions — recognising, sketching and interpreting the graphs of y = sin x, y = cos x and y = tan x for 0° ⩽ x ⩽ 360°; solving trigonometric equations involving sin x, cos x or tan x for 0° ⩽ x ⩽ 360°
• E6.5 Non-right-angled triangles — using the sine and cosine rules in calculations involving lengths and angles for any triangle, and using the formula area of triangle = ½ ab sin C; includes problems involving obtuse angles and the ambiguous case; the sine rule, cosine rule and area formula are given in the List of formulas provided in the exam
• E6.6 Pythagoras’ theorem and trigonometry in 3D — carrying out calculations and solving problems in three dimensions using Pythagoras’ theorem and trigonometry, including calculating the angle between a line and a plane

How to approach it

Before reaching for a formula, identify what kind of triangle the question gives you: a right angle means Pythagoras or SOH CAH TOA is enough (Core and Extended alike), while a triangle with no right angle signals that only the Extended sine or cosine rule will work. Choosing between the sine and cosine rules comes down to what you already know: use the cosine rule when you have three sides, or two sides and the included angle; use the sine rule when you have a side and its opposite angle, plus one more piece of information. Elevation and depression problems (Extended, E6.2) are just right-angled triangle trigonometry with the angle measured from a horizontal line, so redraw them with the horizontal marked before choosing a ratio. For 3D problems, always start by sketching the single 2D right-angled triangle hidden inside the solid — usually formed by a diagonal, a height, and a base length — since the calculation itself is then identical to a 2D question.

Worked example: cosine rule with an obtuse angle (Extended)

Triangle ABC has AB = 7 cm, AC = 10 cm and angle BAC = 110°. Find BC.

BC^2 = AB^2 + AC^2 - 2 x AB x AC x cos(BAC)
BC^2 = 7^2 + 10^2 - 2 x 7 x 10 x cos(110 degrees)
BC^2 = 49 + 100 - 140 x (-0.342)
BC^2 = 149 + 47.9
BC^2 = 196.9
BC = 14.0 cm (3 s.f.)

Because the given angle (110°) is obtuse, cos(110°) is negative, so subtracting a negative term actually increases the result — a detail worth checking mentally before accepting an answer, since it is easy to lose the sign when substituting into a calculator.

Common mistakes

Using SOH CAH TOA or Pythagoras on a triangle that has no right angle, when the sine or cosine rule is required instead. Forgetting that a calculator must be in degree mode, producing wildly incorrect values for sine, cosine or tangent. Mixing up which pair of sides and angle the sine rule needs (a side and its opposite angle) with what the cosine rule needs (three sides, or two sides and the included angle). In 3D problems, attempting to calculate an angle using a 2D face of the solid rather than isolating the actual right-angled triangle formed by the diagonal in question. Losing track of the negative sign that appears when using the cosine rule with an obtuse angle.

Quick revision checklist

• Know SOH CAH TOA and Pythagoras’ theorem for right-angled triangles (Core and Extended).
• Know the exact trigonometric values for 0°, 30°, 45°, 60° and 90° (Extended).
• Practise sketching and reading y = sin x, y = cos x and y = tan x over 0°–360°, and solving trig equations from them (Extended).
• Know when to use the sine rule versus the cosine rule, and be comfortable with the ambiguous case and obtuse angles (Extended).
• Practise isolating the single right-angled triangle inside a 3D solid before calculating (Extended).

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics (0580) syllabus for examination in 2025, 2026 and 2027: official syllabus PDF, Subject content, section 6 “Trigonometry”. Verified 2026-09-06.