the graphs of the trigonometric functions

Trace the height of a point as it travels round and round the unit circle, and plot that height against the angle: you get the rolling wave of the sine graph. The graphs of the trig functions turn the circular motion into a picture you can read, and that picture is the language of every oscillation in science.

The graph of y = sin theta is a smooth wave that starts at 0, rises to 1 at 90 degrees, returns to 0 at 180 degrees, dips to -1 at 270 degrees, and is back to 0 at 360 degrees — then repeats forever. y = cos theta is the identical wave shifted left by 90 degrees, starting at its peak of 1. Both are periodic with period 360 degrees (or 2 x pi radians) and have amplitude 1 (they swing one unit above and below the centre line). A general sinusoid y = A sin(B theta - C) + D bundles four controls: A is the amplitude (height of the swing), the period is 360/B (how fast it repeats), C/B is the phase shift (how far it slides sideways), and D is the vertical shift (the centre line). Tangent's graph is different in character — it climbs from negative infinity to positive infinity over each 180-degree stretch and has vertical asymptotes where cos theta = 0, because tan is sin/cos.

Because these curves repeat, they model anything cyclic: sound, light, alternating current, tides, the swing of a pendulum, the seasons. Reading amplitude, period, and phase shift off a graph — or building a graph to match a real oscillation — is the bread and butter of physics and engineering. A frequent confusion is sign and direction: increasing B squeezes the wave (shorter period), and the phase term subtracts inside, so y = sin(theta - 30 degrees) shifts the curve to the right, not left.