Trigonometric Graphs & Equations

What are trig graphs, really?

Up until now you've probably only used sin, cos and tan for right-angled triangles, where the angle has to be somewhere between 0° and 90° (it can't be obtuse or you don't have a triangle anymore!). Trig graphs free the angle from that restriction. Here, x is just a number that represents an angle — and it can be obtuse (90°–180°), reflex (180°–360°), or even negative or bigger than 360°. The graphs show you what sin, cos and tan do across the whole range of angles, not just the acute ones.

The key thing that makes these graphs special is that they're periodic — the shape repeats itself forever in both directions. Think of a running track: once you've gone all the way round, you're back exactly where you started, and the next lap looks identical to the last one. That's exactly what sin and cos do every 360°.

The graph of y = sin x

y = sin x is a smooth wave. It passes through the origin (0, 0), rises to a maximum height of 1 at x = 90°, comes back down through 0 at x = 180°, dips to a minimum of −1 at x = 270°, and returns to 0 at x = 360° — then the whole pattern repeats.

The graph of y = cos x

y = cos x is exactly the same wave shape as sin x — same amplitude, same period — just shifted. Instead of starting at 0, it starts at its maximum: the y-intercept is (0, 1). It dips to −1 at x = 180°, and comes back up to 1 at x = 360°.

cos x is just sin x shifted 90° to the left.

The graph of y = tan x

asymptotes