Properties of Real Cosine Function
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Theorem
Let $x \in \R$ be a real number.
Let $\cos x$ be the cosine of $x$.
Then:
Cosine Function is Continuous
- $\cos x$ is continuous on $\R$.
Cosine Function is Absolutely Convergent
- $\cos x$ is absolutely convergent for all $x \in \R$.
Cosine of Zero is One
- $\cos 0 = 1$
Cosine Function is Even
- $\map \cos {-z} = \cos z$
That is, the cosine function is even.
Cosine of Multiple of Pi
- $\forall n \in \Z: \cos n \pi = \paren {-1}^n$
Cosine of Half-Integer Multiple of Pi
- $\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$
Shape of Cosine Function
The cosine function is:
- $(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$
- $(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$
- $(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
- $(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$