The Graph of Sine
Simple Explanation
y = sin x is a smooth, periodic wave that oscillates between β1 and 1, passing through the origin, and repeating its exact shape every 2Ο (a full "cycle").
Why Do We Need It?
The sine graph is the archetypal periodic wave β the model for countless real oscillating phenomena, from sound waves to alternating current.
See It
A smooth wave oscillating between -1 and 1, passing through the origin, repeating every 2 pi
Formula
Properties of y = sin x
Domain: all real numbers. Range: [β1, 1]. Period: 2Ο
y=sinx oscillates smoothly between β1 and 1, repeating its exact shape every 2Ο, and passes through the origin.
- x
- β the input angle, in radians
When to use it: Whenever identifying or sketching the basic sine graph and its key features.
Worked Example
Use symmetry to evaluate sine at two related angles
Find sin(Ο/6) and sin(5Ο/6), and note what this tells us about the graph's symmetry.
Why Does This Work?
This symmetry comes directly from the general trig ratio definition on the unit circle: the points at angles ΞΈ and ΟβΞΈ are mirror images across the vertical axis of the circle, and both have the exact same y-coordinate (sine value), even though their x-coordinates (cosine values) differ in sign.
Real-Life Example
Modeling a sound wave
A pure musical tone can be modeled as air pressure oscillating smoothly over time.
The sine graph is exactly the mathematical shape used to model this smooth, repeating oscillation.
Practice
Find sin(Ο/2).
EasyCommon mistake
Assuming sine repeats every Ο instead of the correct 2Ο β that shorter period actually belongs to tangent.
Quick Review
- y=sinx: domain all reals, range [β1,1], period 2Ο.
- Passes through the origin, increasing there.
- sin(ΟβΞΈ) = sin(ΞΈ) β a key symmetry.