Properties of Sine Function

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Theorem

Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:

Sine Function is Continuous

$\sin x$ is continuous on $\R$.

Sine Function is Absolutely Convergent

$\sin x$ is absolutely convergent for all $x \in \R$.

Sine of Zero is Zero

$\sin 0 = 0$

Sine Function is Odd

$\sin \left({-x}\right) = -\sin x$

That is, the sine function is odd.

Sine of Multiple of Pi

$\forall n \in \Z: \sin n \pi = 0$

Sine of Half-Integer Multiple of Pi

$\forall n \in \Z: \sin \left({n + \dfrac 1 2}\right) \pi = \left({-1}\right)^n$

Shape of Sine Function

The sine function is:

$(1): \quad$ strictly increasing on the interval $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$
$(2): \quad$ strictly decreasing on the interval $\left[{\dfrac \pi 2 \,.\,.\, \dfrac {3 \pi} 2}\right]$
$(3): \quad$ concave on the interval $\left[{0 \,.\,.\, \pi}\right]$
$(4): \quad$ convex on the interval $\left[{\pi \,.\,.\, 2 \pi}\right]$