Spin a point around the unit circle (a circle with radius 1) through angle θ, and that point's y-coordinate (sin θ) keeps rising and falling. Stretch out the heights this point has passed through, in time order, sideways, and a smooth, rippling wave-shaped curve appears. This is exactly the graph of the sine function y=sin x. The motion of spinning around a circle and the rippling graph are, in fact, the same thing shown two different ways.
The cosine function y=cos x is built by the exact same principle — the only difference is its starting point. sin starts at 0 when the angle is 0°, but cos already starts at 1 (its highest point) at 0°. That's why the cos graph is exactly the sin graph shifted 90° to the left. Both graphs repeat the exact same shape every 360° (one full turn), and this property — repeating the same shape at regular intervals — is called "periodicity."
There are four dials that let you freely reshape this wave graph. The amplitude (a) determines how high the wave swings, and the period (shorter as b gets larger) determines how much horizontal distance one swing takes. The phase shift (h) pushes the wave left or right, and the vertical shift (d) pushes the entire wave up or down. The pitch (frequency) and volume (loudness) of the sound coming out of a speaker ultimately work on this exact same principle of adjusting amplitude and period.
The tangent function y=tan x looks quite different from sin and cos. It's defined as tan θ = sin θ ÷ cos θ, and at the points where cos θ becomes 0 (90°, 270°...), the denominator becomes 0, so the graph shoots up to the sky and then drops straight down to the floor, breaking apart. A vertical line where the graph is undefined and breaks apart like this is called an "asymptote," and around these asymptotes, the tan graph repeats the same shape every 180° (half the period of sin and cos).
Graphs of trig functions are extremely useful for describing repeating phenomena all around us. The tide coming in and going out twice a day, the waveform of sound coming out of a speaker, the changing height of a car as a Ferris wheel turns, even the strength of the alternating current (AC) flowing through your home's outlets — all of these can be accurately represented with sine and cosine curves. You could fairly say that "anything that rises and falls repeatedly by a fixed rule" is, in effect, described by a trig function graph.
When studying this with kids, it helps to first have them picture a real up-and-down motion, like a Ferris wheel or a swing, and then have them predict what shape appears when you plot that height against time. On our activity page, you can pick one of sin, cos, or tan, change the amplitude, period, phase shift, and vertical shift with sliders, and watch directly how the base wave (dashed) differs from the transformed wave (solid).