# 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 $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$