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Period and amplitude

A sine function oscillates about an equilibrium line. The amplitude is the distance from the equilibrium to a maximum, and it is half the distance from the minimum to the maximum. The period is the length of one oscillation. If there is a coefficient in front of $x$ , the period is $2\pi$ divided by that coefficient. The equilibrium line is the number added outside the sine.

f(x) = A\sin(cx) + d

sine function

p = \frac{2\pi}{c}

the period

Symbols

$A$	the amplitude
$c$	the coefficient of x
$d$	the equilibrium line
$p$	the period

Example

$f(x) = 5\sin(2x) + 1$ has amplitude $5$ and period $2\pi/2 \approx 3.14$ . A maximum of $8$ and a minimum of $2$ give amplitude $(8 - 2)/2 = 3$ .

The period becomes shorter when the coefficient of

x

becomes larger.

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Part of Mathematics R2: Trigonometric functions.