# Properties of Real Sine Function

## Theorem

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

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

Then:

### Real 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

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

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 $\closedint \pi {2 \pi}$