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.

Related Topics

The Unit CircleQuadratic EquationsSlope of a Line

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.