You can already evaluate \(\sin x\), \(\cos x\) and \(\tan x\) for any angle. Plotting these values against \(x\) produces curves that repeat the same pattern over and over — periodic graphs. Three words describe every periodic graph.
Key features of a periodic graph
| Feature | Meaning |
|---|---|
| Midline | Horizontal line midway between the maximum and minimum points. |
| Amplitude | Vertical distance from the midline to a maximum or minimum point. |
| Period | Length of the interval over which the curve repeats itself. |
Values at the quadrant angles
| Function | \(0^\circ\) (\(0\)) | \(90^\circ\) (\(\frac{\pi}{2}\)) | \(180^\circ\) (\(\pi\)) | \(270^\circ\) (\(\frac{3\pi}{2}\)) | \(360^\circ\) (\(2\pi\)) |
|---|---|---|---|---|---|
| \(\sin x\) | \(0\) | \(1\) | \(0\) | \(-1\) | \(0\) |
| \(\cos x\) | \(1\) | \(0\) | \(-1\) | \(0\) | \(1\) |
| \(\tan x\) | \(0\) | undefined | \(0\) | undefined | \(0\) |
\(y=\sin x\): period \(360^\circ\) (\(2\pi\)), amplitude \(1\), \(-1\le\sin x\le 1\). Starts at \(0\) on the \(y\)-axis.
\(y=\cos x\): period \(360^\circ\) (\(2\pi\)), amplitude \(1\), \(-1\le\cos x\le 1\). Starts at \(1\) on the \(y\)-axis.
\(y=\tan x\): period \(180^\circ\) (\(\pi\)), no amplitude, all real values. Vertical asymptotes at \(x=90^\circ\) and \(x=270^\circ\).
Each graph continues in both directions for all values of \(x\): every feature you see in one period repeats in every other period.
\(y=\tan x\) has no maximum, no minimum and no amplitude. Its graph approaches the asymptotes but never touches them — show asymptotes as dashed vertical lines in a sketch.
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