Periodicity of Trigonometric Functions
Sine and cosine are periodic with least period , where is the first positive zero of ; arc length gives .
There is a smallest positive such that , and
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are periodic with period , i.e.
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for all .
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for all .
Pi and the unit circle
is the perimeter of the unit circle,
Proof
On the terms of the sine series alternate in pairs of decreasing size, so : strictly decreases and has at most one zero there. Grouping terms shows and , so by the Intermediate Value Theorem vanishes at some . There , and positivity of on selects . All shift identities follow from the addition formulae for and .
For the circle, maps bijectively onto . Given with , strict monotonicity of on produces with , and or according to the sign of , so is surjective; injectivity uses that the least positive period is . Hence
Related
Stated in
- Proposition 5.24 (Periodicity of Trignometric Functions)§5.5 Trigonometric Functions
- Lemma 5.26§5.5 Trigonometric Functions