What is Trigonometry?

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. Derived from the Greek words trigonon (triangle) and metron (measurement), trigonometry forms a cornerstone of GCSE and A-Level maths, as well as being fundamental to physics, engineering, navigation, and architecture. The subject revolves around six functions — sine, cosine, tangent, cosecant, secant, and cotangent — each of which describes a specific ratio between the sides of a right-angled triangle or, more broadly, the coordinates of a point on a circle.

At GCSE level (AQA, Edexcel, OCR), students are expected to use SOHCAHTOA to find missing sides and angles in right-angled triangles. At A-Level, the subject expands significantly to include the sine rule, cosine rule, trigonometric identities, inverse functions, general solutions, and calculus applied to trig functions. This calculator covers all the core needs across both levels.

SOHCAHTOA Explained

SOHCAHTOA is the most widely used mnemonic in GCSE maths. It encodes the three primary trigonometric ratios:

To identify the sides, always label them relative to the angle you are working with. The hypotenuse is always the longest side, directly opposite the right angle. The opposite side is the one directly opposite your angle, and the adjacent side is the one next to your angle (that is not the hypotenuse).

Example: In a right triangle where angle A = 35° and the hypotenuse = 10 cm, find the opposite side:
sin 35° = opposite / 10
opposite = 10 × sin 35° = 10 × 0.5736 = 5.74 cm

The Six Trigonometric Functions

Beyond the primary three ratios, there are three reciprocal functions:

The reciprocal functions are particularly important at A-Level, where you must know their graphs, domains, ranges, and identities.

Degrees and Radians

Angles can be measured in degrees or radians. Degrees are the everyday unit (a full circle = 360°). Radians are the mathematically natural unit (a full circle = 2π radians), and they become essential for calculus — the derivatives of sin and cos only work out neatly when you use radians.

To convert: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.

Key values to memorise: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

Gradians are a third unit used in some surveying and European engineering contexts: a full circle = 400 gradians. A right angle = 100 gradians. You can convert with: 1 gradian = 0.9° = π/200 radians.

The Unit Circle

The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured anticlockwise from the positive x-axis, the point on the circle has coordinates (cos θ, sin θ). This gives a definition of sine and cosine that works for any angle, not just those between 0° and 90°. From the unit circle:

• In the first quadrant (0° to 90°): sin, cos, and tan are all positive.
• In the second quadrant (90° to 180°): only sin is positive.
• In the third quadrant (180° to 270°): only tan is positive.
• In the fourth quadrant (270° to 360°): only cos is positive.

The CAST diagram (from bottom-right going anticlockwise: C, A, S, T) captures which functions are positive in each quadrant. At A-Level, this is used to find all solutions to trig equations in a given range.

Inverse Trigonometric Functions

Inverse trig functions find the angle when you know the ratio. They are written as sin¹, cos¹, tan¹ or arcsin, arccos, arctan:

• arcsin(x): returns angle in −90° to 90° (principal value) given sin θ = x, where x ∈ [−1, 1]
• arccos(x): returns angle in 0° to 180° given cos θ = x, where x ∈ [−1, 1]
• arctan(x): returns angle in −90° to 90° given tan θ = x, for any real x

At GCSE you use these to find angles in right triangles: if sin A = 0.6, then A = arcsin(0.6) = 36.87°. At A-Level you must consider general solutions: for sin θ = 0.5, the general solution is θ = 30° + 360n° or θ = 150° + 360n° for any integer n.

Trigonometric Identities

Trig identities are equations that hold for all valid angles. Knowing them allows you to simplify expressions and solve equations. The core identities at A-Level are:

Graphs of sin, cos, and tan

Understanding the graphs of trig functions is essential for A-Level maths:

• y = sin x: Oscillates between −1 and 1. Period = 360° (2π rad). Starts at (0,0), rises to peak at 90°, back to 0 at 180°, trough at 270°, back to 0 at 360°.
• y = cos x: Oscillates between −1 and 1. Period = 360°. Starts at (0,1), identical to sin x but shifted 90° to the left (cos x = sin(x + 90°)).
• y = tan x: Repeating pattern with period 180° (π rad). Has vertical asymptotes at ±90°, ±270° (where cos = 0).

Key transformations: y = a sin(bx + c) + d has amplitude |a|, period 2π/b, phase shift −c/b, and vertical translation d. This is essential for A-Level exam questions on transformations of trig graphs.

Real-World Applications

Trigonometry is not merely an academic exercise — it underpins a vast range of practical applications:

• Navigation and GPS: Ships and aeroplanes use bearings and trigonometry to calculate distances and headings. GPS satellites use triangulation to pinpoint locations.
• Architecture and construction: Roof angles, ramp gradients, and structural load calculations all rely on trig. The pitch of a roof is calculated as the arctan of rise over run.
• Physics — waves and oscillations: Sine waves model sound, light, and electromagnetic radiation. Simple harmonic motion (springs, pendulums) is described by x = A sin(ωt + φ).
• Engineering: AC electrical circuits use phasor diagrams based on sine and cosine. Signal processing uses the Fourier transform, which decomposes any signal into sine waves.
• Astronomy: Astronomers calculate distances to stars using parallax, a method that requires trigonometry applied to very small angles.
• Computer graphics and games: Rotation matrices, 3D rendering, and animation all rely heavily on trig functions to transform coordinates in 2D and 3D space.

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