Form 5 · Chapter 6

Graphs of Sine, Cosine and Tangent Functions

The graphs of sine, cosine and tangent are periodic. Each has a characteristic shape, period and range that we can read off directly.

Sine and cosine graphs

The graphs of $y=\sin x$ and $y=\cos x$ are smooth waves. Over $0^\circ\le x\le 360^\circ$ each completes one full cycle. Their period is $360^\circ$ and their amplitude (the maximum distance from the middle) is $1$. So both range between $-1$ and $1$: the maximum value is $1$ and the minimum is $-1$.

  • $y=\sin x$ passes through $(0^\circ,0)$, peaks at $x=90^\circ$;
  • $y=\cos x$ starts at its maximum $(0^\circ,1)$.

The tangent graph

The graph of $y=\tan x$ is different: it has no maximum or minimum and its period is $180^\circ$. It is undefined at $x=90^\circ$ and $x=270^\circ$, where it has vertical asymptotes.

Key formula

$y=a\sin(bx):\quad \text{amplitude}=|a|,\qquad \text{period}=\frac{360^\circ}{b}$

Example

For $y=3\sin x$ the amplitude is $3$, so the curve runs from $-3$ to $3$; the period is still $360^\circ$. For $y=\sin 2x$ the period is $\dfrac{360^\circ}{2}=180^\circ$.

Remember

Amplitude controls the height; the coefficient of $x$ controls how many cycles fit in $360^\circ$.

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