Properties of Real Sine Function
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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 $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$