I am currently working on a challenge problem where I need to show that there is a point $x \in \mathbb{R_+}$ such that $\cos(x) = 0$ using only a few properties of the cosine function. In particular, the only properties of the cosine function that I can use are:
$\cos(x)$ is continuous
$\cos(x) = Re(\exp(z))$ for $z \in \mathbb{C}$
- $\cos^2(x) + \sin^2(x)=1$
- $\displaystyle \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n!)}x^{2n}$
- $\displaystyle \exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$
My strategy is to use the intermediate value theorem on the interval $[0,a]$ since it's easy to show that $\cos(0) = 1$. If I could show that there is a point $a \in \mathbb{R}_+$ s.t. $\cos(a) < 0$, then the IVT and the continuity of the cosine function would allow me to conclude that there has to be some $x\in [0,a]$ such that $\cos(x) = 0$.