The sine graph
The graph of \(y = \sin x\) is a smooth wave that rolls along the \(x\)-axis forever. Over one full turn from \(0^\circ\) to \(360^\circ\) it does one complete cycle.
- Cycle
- One complete repeat of the wave before the pattern starts again. The width of one cycle along the \(x\)-axis is the period.
Its key points across one cycle: it starts at \((0^\circ, 0)\), climbs to a maximum of \(1\) at \(90^\circ\), returns to \(0\) at \(180^\circ\), drops to a minimum of \(-1\) at \(270^\circ\), and comes back to \(0\) at \(360^\circ\). The output never leaves the band from \(-1\) to \(1\).
The cosine graph
The graph of \(y = \cos x\) has exactly the same wave shape and the same range from \(-1\) to \(1\), but it is shifted: it starts at the top.
Its key points across one cycle: it starts at a maximum of \(1\) at \((0^\circ, 1)\), falls to \(0\) at \(90^\circ\), reaches a minimum of \(-1\) at \(180^\circ\), returns to \(0\) at \(270^\circ\), and is back at \(1\) by \(360^\circ\). In fact the cosine graph is the sine graph slid \(90^\circ\) to the left.
For \(y = \cos x\), find every value of \(x\) between \(0^\circ\) and \(360^\circ\) where \(y = 0\).
The tangent graph
The graph of \(y = \tan x\) breaks the wave pattern. Because \(\tan x = \dfrac{\sin x}{\cos x}\), it shoots off to infinity wherever \(\cos x = 0\), so it is made of separate branches.
- Asymptote
- A vertical line the curve races towards but never touches. For \(y = \tan x\) these sit at \(90^\circ\), \(270^\circ\), and every \(180^\circ\) from there.
Each branch climbs from far below, passes through a zero (at \(0^\circ\), \(180^\circ\), \(360^\circ\)), and rises far above before the next asymptote. Unlike sine and cosine, tangent has no maximum or minimum: its output is unbounded. It repeats every \(180^\circ\), not \(360^\circ\).
Key features to name
When a question asks you to describe or compare these graphs, use this vocabulary.
| Feature | \(y = \sin x\) | \(y = \cos x\) | \(y = \tan x\) |
|---|---|---|---|
| Period | \(360^\circ\) | \(360^\circ\) | \(180^\circ\) |
| Amplitude | \(1\) | \(1\) | none |
| Range | \(-1\) to \(1\) | \(-1\) to \(1\) | all values |
| Starts at | \((0^\circ, 0)\) | \((0^\circ, 1)\) | \((0^\circ, 0)\) |
| Asymptotes | none | none | \(90^\circ, 270^\circ, \ldots\) |
Amplitude is a half-height
The amplitude is the distance from the middle line up to a peak, not the full gap from bottom to top. For \(y = \sin x\) the wave runs from \(-1\) to \(1\), a total height of \(2\), but the amplitude is \(1\).
Where this is assessed
Naming the period, amplitude, range and asymptotes is Criterion A (Knowing and understanding). Sketching a clean, labelled curve with the axes marked is Criterion C (Communicating). Linking the wave to real cycles such as tides, daylight hours and sound is Criterion D (Applying mathematics in real-life contexts).
Check yourself
1. What is the period of \(y = \tan x\)? +
The tangent graph repeats after every half turn, so its period is \(180^\circ\), unlike sine and cosine, which repeat every \(360^\circ\).
2. Give the coordinates of the minimum of \(y = \cos x\) for \(0^\circ \le x \le 360^\circ\). +
Cosine starts at its maximum and reaches its lowest point half a cycle later, at \(180^\circ\), where the value is \(-1\). So the minimum is at \((180^\circ, -1)\).
3. Why does \(y = \tan x\) have a vertical asymptote at \(90^\circ\)? +
Because \(\tan x = \dfrac{\sin x}{\cos x}\), and at \(90^\circ\) the denominator \(\cos 90^\circ = 0\). Dividing by zero is undefined, so the curve has no value there and races off towards the vertical line \(x = 90^\circ\).
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