Cosine definition
Cosine is a trigonometric function that connects an angle with the ratio of two sides in a right-angled triangle. In GCSE maths, cosine is usually introduced through SOHCAHTOA. The cosine ratio is written as cos θ = adjacent ÷ hypotenuse. The angle is usually called θ, which is the Greek letter theta. The adjacent side is the side next to the angle, and the hypotenuse is the longest side opposite the right angle.
For example, imagine a right-angled triangle where the angle is 60°, the adjacent side is 5 cm and the hypotenuse is 10 cm. The cosine of the angle is cos 60° = 5 ÷ 10 = 0.5. This means the adjacent side is half the length of the hypotenuse for that angle. Students can use this ratio to find missing sides or angles when enough information is given.
Cosine is used in trigonometry, geometry, vectors, coordinate geometry, mechanics, waves, graphs and modelling. At GCSE level, cosine helps students solve right-angled triangle problems, including questions involving ladders, ramps, bearings, elevation and construction. At A-Level, cosine appears in radians, trigonometric identities, the cosine graph, compound-angle formulas, differentiation, integration, vectors and harmonic motion.
A simple cosine calculation might ask for a missing adjacent side. If cos 40° = adjacent ÷ 12, then adjacent = 12 × cos 40°. Using a calculator, this gives approximately 9.19. The formula can also be rearranged to find the hypotenuse: hypotenuse = adjacent ÷ cos θ. Knowing how to rearrange the cosine formula is important because exam questions can ask for different missing parts of the triangle.
Cosine is also important beyond right-angled triangles. The cosine rule is used for non-right-angled triangles. It is written as a² = b² + c² − 2bc cos A. This formula is useful when you know two sides and the included angle, or when you know all three sides and need to find an angle. It connects trigonometry with Pythagoras’ theorem and extends triangle solving to more general shapes.
The cosine graph is another important idea. The graph of y = cos x repeats in a wave pattern. Its maximum value is 1, its minimum value is -1, and it repeats every 360° or every 2π radians. This makes cosine useful for modelling repeating behaviour such as tides, sound waves, light waves, circular motion and seasonal patterns.
Understanding cosine is important because many students confuse the trigonometric ratios. A helpful memory is CAH from SOHCAHTOA: Cosine = Adjacent ÷ Hypotenuse. Before using the formula, always identify the angle you are working from. The adjacent side changes depending on which acute angle is chosen, but the hypotenuse is always the longest side opposite the right angle.
Related ideas include sine, tangent, SOHCAHTOA, Pythagoras’ theorem, the cosine rule, radians, trigonometric graphs, bearings and vectors. In exams, check whether your calculator is in degrees or radians. A correct method can produce a wrong answer if the calculator mode is incorrect. A strong definition of cosine helps students understand not just which button to press, but why the ratio works.