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.
Key Formula
sin(θ) = opposite / hypotenuse
LaTeX: \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | angle measured from the positive x-axis | radians or degrees |
| opposite | side of right triangle opposite to angle θ | length units |
| hypotenuse | longest side of right triangle, opposite the right angle | length units |
Worked Example
Problem
A ladder 5 m long leans against a wall making an angle of 30° with the wall. How high up the wall does the ladder reach?
Solution
Step 1: The angle between the ladder and the wall is 30°, so the angle between the ladder and the ground is 60°. Step 2: The height h is the side opposite the 60° angle, with the ladder as hypotenuse. Step 3: sin(60°) = h / 5 → h = 5 × sin(60°) = 5 × (√3/2) = 5 × 0.8660 = 4.330 m.
Answer
The ladder reaches 4.33 m up the wall (to 3 significant figures).
Sine Values at Key Angles
| Angle (degrees) | Angle (radians) | sin(θ) | Exact Value |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | 0.5000 | 1/2 |
| 45° | π/4 | 0.7071 | √2/2 |
| 60° | π/3 | 0.8660 | √3/2 |
| 90° | π/2 | 1.0000 | 1 |
| 180° | π | 0 | 0 |
Interactive Tools
Desmos Graphing Calculator
Plot y = sin(x) and explore amplitude, period, and phase shift dynamically.
Open ToolKhan Academy — Trigonometry
Structured lessons and practice on the sine function, unit circle, and applications.
Open ToolPhET Wave on a String
Interactive simulation illustrating sinusoidal wave motion in a physical context.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
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.
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.
Radian
A radian is the SI unit of angular measure defined as the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. Since the full circumference is 2πr, one complete revolution equals 2π radians, giving the exact conversion 180° = π radians. Radians are the natural unit for trigonometry and calculus because they make derivative formulas for trigonometric functions simple (d/dθ sin θ = cos θ holds only when θ is in radians).
Derived from the Latin sinus ("curve, fold, bay"), a mistranslation of the Arabic jiba (جيب), itself a transliteration of the Sanskrit jyā ("bowstring"). Medieval European scholars misread the Arabic as jaib ("fold of a garment"), which was then rendered as sinus in Latin translations of al-Khwarizmi's work (c. 12th century AD).