Basic Properties of Cosine Function

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

$\cos \left({-x}\right) = \cos x$

That is, the cosine function is even.

Cosine of Multiple of Pi

$\forall n \in \Z: \cos n \pi = \left({-1}\right)^n$

Cosine of Multiple of Pi Plus Half

$\forall n \in \Z: \cos \left({n + \dfrac 1 2}\right) \pi = 0$

Shape of Cosine Function

The cosine function is:

$(1): \quad$ strictly decreasing on the interval $\left[{0 .. \pi}\right]$

$(2): \quad$ strictly increasing on the interval $\left[{\pi .. 2 \pi}\right]$

$(3): \quad$ concave on the interval $\left[{-\dfrac \pi 2 .. \dfrac \pi 2}\right]$

$(4): \quad$ convex on the interval $\left[{\dfrac \pi 2 .. \dfrac {3 \pi} 2}\right]$

Also see

• Basic Properties of Sine Function