Functions
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Sine and Cosine

Explore amplitude, period, and phase shift — one slider at a time

The sine function y = \sin(x) creates a smooth wave that repeats every 2\pi units. It's one of the most important functions in all of mathematics — it describes sound waves, light waves, tides, and anything that oscillates.

In this lesson, you'll explore the general sine function y = a \cdot \sin(bx + c) using three sliders: a controls the amplitude (height), b controls the frequency (how many waves fit), and c controls the phase shift (left/right sliding). A gray reference wave y = \sin(x) stays on screen so you can always compare.

By the end, you'll also see how cosine is just a shifted sine — same wave, different starting point.

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FAQ

What is the sine function?
The sine function y = \sin(x) takes an angle (in radians) and returns a value between -1 and 1. It creates a smooth, repeating wave that crosses zero at 0, π, 2π, etc., reaches a maximum of 1 at π/2, and a minimum of -1 at 3π/2. It's the foundation of trigonometry and wave physics.
What are amplitude, period, and phase shift?
For y = a \sin(bx + c): Amplitude = |a|, the height of the wave from center to peak. Period = 2π/|b|, the horizontal distance for one full cycle. Phase shift = -c/b, how far the wave slides left or right. These three parameters control every aspect of the wave's shape and position.
What is the difference between sine and cosine?
Cosine is just sine shifted left by π/2: \cos(x) = \sin(x + \pi/2). They have the same wave shape, same amplitude, same period — the only difference is where they start. Sine starts at 0, cosine starts at 1. On the unit circle, sine is the y-coordinate and cosine is the x-coordinate.
How is sine connected to the unit circle?
On the unit circle, sine is the y-coordinate and cosine is the x-coordinate of a point at angle θ. As θ increases from 0 to 2π, the y-coordinate traces out the sine wave. That's why sin(0) = 0, sin(π/2) = 1, sin(π) = 0, and sin(3π/2) = -1.

Related Topics

The Unit CircleQuadratic EquationsSlope of a Line