Cosine Function
The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle, and extended to all real numbers as the x-coordinate of a point on the unit circle at angle θ from the positive x-axis. Like sine, it is periodic with period 2π and range [−1, 1], but is phase-shifted by π/2 relative to sine (cos θ = sin(θ + π/2)). Cosine is widely used in Fourier analysis, wave optics, mechanical vibrations, and calculating dot products of vectors.
Key Formula
cos(θ) = adjacent / hypotenuse
LaTeX: \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | angle measured from the positive x-axis | radians or degrees |
| adjacent | side of right triangle adjacent to angle θ (not the hypotenuse) | length units |
| hypotenuse | longest side of right triangle | length units |
Worked Example
Problem
A ramp makes an angle of 25° with the horizontal. If the ramp is 8 m long, what is the horizontal distance it covers?
Solution
Step 1: Identify the relevant ratio: horizontal distance = adjacent side, ramp length = hypotenuse. Step 2: cos(25°) = adjacent / 8 → adjacent = 8 × cos(25°). Step 3: cos(25°) ≈ 0.9063. Step 4: adjacent = 8 × 0.9063 = 7.250 m.
Answer
The ramp covers a horizontal distance of approximately 7.25 m.
Cosine Values at Key Angles
| Angle (degrees) | Angle (radians) | cos(θ) | Exact Value |
|---|---|---|---|
| 0° | 0 | 1.0000 | 1 |
| 30° | π/6 | 0.8660 | √3/2 |
| 45° | π/4 | 0.7071 | √2/2 |
| 60° | π/3 | 0.5000 | 1/2 |
| 90° | π/2 | 0 | 0 |
| 180° | π | −1.0000 | −1 |
Interactive Tools
Desmos Graphing Calculator
Visualise the cosine curve and compare it to sine to see the phase relationship.
Open ToolKhan Academy — Trigonometry
Comprehensive tutorials on cosine, the unit circle, and inverse cosine.
Open ToolWolfram Alpha
Evaluate cosine for any angle, compute identities, and plot the function symbolically.
Open Tool
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Related Terms
Sine Function
The sine function is a fundamental trigonometric function defined for an angle θ in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse, extended to all real numbers via the unit circle. It is a periodic function with period 2π, amplitude 1, and range [−1, 1], producing a smooth oscillating wave. Sine is essential in modelling wave phenomena including sound, light, alternating current, and simple harmonic motion.
Tangent Function
The tangent function is defined as the ratio of the sine to the cosine of an angle (tan θ = sin θ / cos θ), or equivalently as the ratio of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, the tangent function has a period of π and is undefined at θ = π/2 + nπ (where n is any integer) because cosine equals zero at those points, producing vertical asymptotes. Tangent is fundamental in calculating slopes of lines, angles of elevation and depression, and in integral calculus substitutions.
Law of Cosines
The Law of Cosines generalises the Pythagorean theorem to any triangle, relating the square of one side to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. When the included angle is 90°, cos(90°) = 0 and the formula reduces to the Pythagorean theorem a² = b² + c². It is used to solve triangles when three sides (SSS) or two sides and the included angle (SAS) are known, and is fundamental in physics (vector addition), engineering, and 3D geometry.
A contraction of the Latin complementi sinus ("sine of the complement"), because cos(θ) = sin(90° − θ). The abbreviation was popularised by English mathematician Edmund Gunter in his 1620 work Canon triangulorum, where he introduced the notation "co.sinus" alongside other complementary trig functions.