Properties of Real Sine Function

Theorem

Let $\sin: \R \to \R$ denote the real sine function.

Then:

Real Sine Function is Continuous

The real sine function $\sin: \R \to \R$ is continuous on $\R$.

Sine Function is Absolutely Convergent

The real sine function $\sin: \R \to \R$ is absolutely convergent.

Sine of Zero is Zero

$\sin 0 = 0$

Sine Function is Odd

$\map \sin {-z} = -\sin z$

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

$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$

Shape of Sine Function

The sine function is:

$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$

$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

$(3): \quad$ concave on the interval $\closedint 0 \pi$

$(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$

Also see

• Properties of Real Cosine Function