Trigonometry definition
Trigonometry is the part of mathematics that studies the relationship between angles and side lengths. It is most commonly introduced through right-angled triangles, where the three main trigonometric ratios are sine, cosine and tangent. These ratios allow students to find missing side lengths or missing angles when enough information is known about a triangle.
In a right-angled triangle, the hypotenuse is the longest side and is opposite the right angle. The opposite side is across from the angle being used, and the adjacent side is next to the angle being used but is not the hypotenuse. These labels are important because sine, cosine and tangent each use different side pairs. The labels can change depending on which angle is chosen.
Trigonometry is used in geometry, bearings, maps, architecture, construction, engineering, physics, navigation, computer graphics, waves and real-life measurement. At GCSE level, students use trigonometry to calculate missing lengths and angles in right-angled triangles. At A-Level, trigonometry expands to include graphs, identities, exact values, radians, equations, calculus and periodic modelling.
The memory aid SOHCAHTOA is often used for right-angled trigonometry. SOH means sine equals opposite divided by hypotenuse. CAH means cosine equals adjacent divided by hypotenuse. TOA means tangent equals opposite divided by adjacent. This helps students choose the correct ratio after labelling the sides of the triangle from the angle in the question.
For example, if a right-angled triangle has an angle of 30 degrees and a hypotenuse of 10 cm, the opposite side can be found using sine. Since sine of 30 degrees is 0.5, the opposite side is 0.5 times 10, which equals 5 cm. This shows how trigonometry turns information about an angle into information about a length.
Trigonometry can also be used to find missing angles. If two side lengths are known, students choose sine, cosine or tangent depending on which sides are involved, calculate the ratio, and then use the inverse trigonometric function on a calculator. For example, if the opposite and adjacent sides are known, inverse tangent can be used to find the angle.
Beyond right-angled triangles, trigonometry includes sine and cosine rules, area of a triangle, trigonometric graphs and exact trigonometric values. The sine and cosine graphs repeat in wave-like patterns, which makes them useful for modelling sound, light, tides, alternating current and circular motion. This is why trigonometry is important in both pure maths and applied science.
Related ideas include sine, cosine, tangent, SOHCAHTOA, right-angled triangles, Pythagoras’ theorem, bearings, sine rule, cosine rule and trigonometric graphs. A clear definition of trigonometry helps students understand that the topic is about relationships between angles and lengths. In exams, label the triangle carefully, choose the correct ratio and make sure the calculator is in the correct angle mode.