The sine and cosine waves
Both \(y=\sin x\) and \(y=\cos x\) are smooth waves that rise and fall between \(-1\) and \(+1\), repeating every \(360^\circ\). They are actually the same wave, just shifted along: cosine is sine started a quarter-turn early.
| \(x\) | \(0^\circ\) | \(90^\circ\) | \(180^\circ\) | \(270^\circ\) | \(360^\circ\) |
|---|---|---|---|---|---|
| \(\sin x\) | \(0\) | \(1\) | \(0\) | \(-1\) | \(0\) |
| \(\cos x\) | \(1\) | \(0\) | \(-1\) | \(0\) | \(1\) |
So \(y=\sin x\) starts at the origin, climbs to a peak at \(90^\circ\), and crosses back down. \(y=\cos x\) starts at its peak of 1 and falls. Both take the value 0, plus and minus 1 at predictable quarter-turns, which is all you need to sketch them.
The tangent graph
The tangent graph looks nothing like the other two. Because \(\tan x = \dfrac{\sin x}{\cos x}\), it shoots off to infinity wherever \(\cos x = 0\). That gives a repeating shape with vertical asymptotes at \(90^\circ, 270^\circ\) and every \(180^\circ\) from there.
- It repeats every \(180^\circ\), not \(360^\circ\), so its period is half that of sine and cosine.
- It has no maximum or minimum: the output runs from minus infinity to plus infinity, so amplitude has no meaning for tan.
- It passes through the origin, climbing steeply through \(0^\circ, 180^\circ, 360^\circ\).
Asymptotes are gaps, not lines
Draw the asymptotes of \(y=\tan x\) as dashed lines and make sure the curve never touches them. \(\tan 90^\circ\) is undefined, the graph has a break there.
Amplitude, period and phase
A general sine or cosine curve is written in the form \(y = a\sin\big(b(x - c)\big) + d\). Each letter controls one transformation:
- Amplitude \(|a|\)
- Half the distance from the highest point to the lowest. It stretches the wave vertically, so the peaks reach \(+a\) and the troughs reach \(-a\) (before any vertical shift).
- Period
- The horizontal length of one full repeat. For sine and cosine it is \(\dfrac{360^\circ}{b}\); the larger \(b\) is, the more squashed the wave.
- Phase shift \(c\)
- A horizontal slide. A positive \(c\) moves the whole curve \(c\) to the right. The \(+d\) at the end slides it up or down instead.
Reading it off the equation
With the form fixed, sketching a transformed curve is just reading off the numbers.
State the amplitude, period and phase shift of \(y = 4\sin(3x - 60^\circ)\).
Where this is assessed
Identifying transformations from an equation is Criterion A, and a labelled sketch with the correct scale on both axes is Criterion C. Always mark the period and the max and min values on your sketch.
Check yourself
1. What is the period of \(y = \cos(4x)\)? +
Period \(= \dfrac{360^\circ}{b}\) with \(b=4\), so \(\dfrac{360^\circ}{4} = \mathbf{90^\circ}\).
2. For \(y = 5\sin x - 2\), state the amplitude and the maximum value. +
The amplitude is \(|a| = \mathbf{5}\). The wave peaks at \(\sin x = 1\), giving a maximum of \(5(1) - 2 = \mathbf{3}\) (and a minimum of \(5(-1)-2 = -7\)).
3. Why does \(y = \tan x\) have vertical asymptotes? +
Because \(\tan x = \dfrac{\sin x}{\cos x}\), and wherever \(\cos x = 0\) you would be dividing by zero, which is undefined. This happens at \(90^\circ, 270^\circ\) and every \(180^\circ\), so the graph has an asymptote at each of those values.
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