Can we solve this equation $\frac{\cos\theta}{\cos{\theta}^2}=k$

Question

I was in doubt that we can solve these type of Equation or not: $\frac{\cos\theta}{\cos{\theta}^2}=k$ where $k$ is a given constant.

Answer 1

This is typically an equation you cannot solve analytically. Only numerical methods would lead to a solution for the more general situation. In order to make life simpler, let us write $$f(\theta)=\cos(\theta)-k\cos(\theta^2)$$ A classical and simple method is Newton iterative scheme which starting with a guess $\theta_0$ will udpate according to $$\theta_{n+1}=\theta_{n}-\frac {f(\theta_n)} {f'(\theta_n)}$$ For illustration purposes, let us assume $k=10$ and search for the first positive solution for which we shall use $\theta_0=1$. The procedure will then provide the following iterates : $1.30415$, $1.24150$, $1.24030$ which is the solution.