Tangent Function is Periodic on Reals
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Theorem
The real tangent function is periodic with period $\pi$.
This can be written:
- $\tan x = \map \tan {x \bmod \pi}$
where $x \bmod \pi$ denotes the modulo operation.
Proof
\(\ds \map \tan {x + \pi}\) | \(=\) | \(\ds \frac {\map \sin {x + \pi} } {\map \cos {x + \pi} }\) | Definition of Real Tangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\sin x} {-\cos x}\) | Sine and Cosine are Periodic on Reals | |||||||||||
\(\ds \) | \(=\) | \(\ds \tan x\) |
From Derivative of Tangent Function, we have that:
- $\map {D_x} {\tan x} = \dfrac 1 {\cos^2 x}$
provided $\cos x \ne 0$.
From Shape of Cosine Function, we have that $\cos > 0$ on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From Derivative of Monotone Function, $\tan x$ is strictly increasing on that interval, and hence can not have a period of less than $\pi$.
Hence the result.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.5 \ (2)$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Shifts and periodicity
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Shifts and periodicity