Sine and Cosine are Periodic on Reals

Theorem

The real sine function and real cosine function are periodic on the set of real numbers $\R$:

Real Cosine Function is Periodic

$\exists L \in \R_{\neq 0}: \forall x \in \R: \cos x = \map \cos {x + L}$

Real Sine Function is Periodic

The real sine function is periodic with the same period as the real cosine function.

Pi

The real number $\pi$ (called pi, pronounced pie) is uniquely defined as:

$\pi := \dfrac p 2$

where $p \in \R$ is the period of $\sin$ and $\cos$.

Cosine of Angle plus Straight Angle

$\map \cos {x + \pi} = -\cos x$

Sine of Angle plus Straight Angle

$\map \sin {x + \pi} = -\sin x$

Sign of Cosine on $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

$\cos x$ is strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

Sign of Sine on $\openint 0 \pi$ and $\openint \pi {2 \pi}$

$\sin x$ is strictly positive on the interval $\openint 0 \pi$ and strictly negative on the interval $\openint \pi {2 \pi}$

Zeroes of Cosine

$\cos x = 0$ if and only if $x = \paren {n + \dfrac 1 2} \pi$ for some $n \in \Z$.

Zeroes of Sine

$\sin x = 0$, if and only if $x = n \pi$ for some $n \in \Z$.

Note

Given that we have defined sine and cosine in terms of a power series, it is a plausible proposition to define $\pi$ using the same language.

$\pi$ is, of course, the famous irrational constant $3.14159 \ldots$.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.4$