Variations and table of values

Analysing a periodic function

Thanks to the period 2π and the parity symmetries, it is sufficient to analyse the sine and cosine functions over an interval such as [0; π], and then extend the results by symmetry to understand their behaviour throughout R.

Table of variations of sine on [0; π]

x	0	π/2	π
sin(x)	0	1 (maximum)	0

sin increases on [0; π/2] and then decreases on [π/2; π].

Table showing the behaviour of cosine on [0; π]

x	0	π
cos(x)	1 (maximum)	-1 (minimum)

cos decreses over the entire interval [0; π].

On the entire real line

Using periodicity (2π) and parity, sine and cosine alternate indefinitely between increasing and decreasing phases: they are NEVER either increasing or decreasing over the entire real line.

Example

To compare sin(2) and sin(3), we place 2 and 3 on the unit circle: π/2 ≈ 1.57 and π ≈ 3.14, so 2 and 3 are both in [π/2; π], an interval where sin is decreasing. As 2 < 3, we conclude that sin(2) > sin(3).

Common pitfall

Never simply say ‘sin is increasing’: always specify the interval. A non-constant periodic function cannot be monotonic over the entire real line.