Tangent definition
Tangent is one of the three main trigonometric ratios used with right-angled triangles. It connects an angle to two side lengths: the side opposite the angle and the side adjacent to the angle. The opposite side is across from the angle being used. The adjacent side is next to the angle and is not the hypotenuse. Tangent helps link angles and lengths in a precise way.
In a right-angled triangle, tangent of an angle is found by dividing the opposite side by the adjacent side. This is remembered using the TOA part of SOHCAHTOA. TOA means Tangent equals Opposite divided by Adjacent. For example, if the opposite side is 6 cm and the adjacent side is 3 cm, the tangent of the angle is 6 divided by 3, which equals 2.
Tangent is used in trigonometry, geometry, bearings, gradients, vectors, physics, engineering, construction, navigation and real-life measurement. At GCSE level, students use tangent to find missing sides and angles in right-angled triangles. At A-Level, tangent appears in trigonometric graphs, identities, equations, calculus, radians, circular motion and modelling periodic behaviour.
To use tangent correctly, first choose the angle in the triangle. Then label the opposite and adjacent sides from that angle. The hypotenuse is not used in the tangent ratio. This is one reason students must label carefully. If a different angle is chosen, the opposite and adjacent sides may change, even though the triangle is the same.
Tangent can be used to find a missing side. If an angle is 40 degrees and the adjacent side is 10 cm, the opposite side can be found using opposite equals tangent of 40 degrees times 10. The calculator gives the tangent value, and multiplying by the adjacent side gives the missing length. This method is common in height, distance and bearings questions.
Tangent can also be used to find an angle. If the opposite side and adjacent side are known, divide the opposite by the adjacent and then use the inverse tangent function on a calculator. For example, if the opposite side is 5 and the adjacent side is 5, the tangent value is 1. The angle with tangent 1 in a right-angled triangle is 45 degrees.
Tangent also has a wider meaning in graph work and calculus. A tangent line touches a curve at a point and has the same gradient as the curve at that point. This meaning is connected to rates of change and differentiation. Although the triangle ratio and tangent line are different ideas, both are important in advanced maths and both connect angles, gradients and change.
Related ideas include trigonometry, SOHCAHTOA, opposite side, adjacent side, sine, cosine, Pythagoras’ theorem, bearings, gradients, tangent lines and trigonometric graphs. A clear definition of tangent helps students know which sides are needed and when the ratio applies. In exams, check that the triangle is right-angled before using basic SOHCAHTOA methods and make sure the calculator is in the correct angle mode.