Finding the intersection of a line and standard wave function Let's say I have two functions $f(x)=5\cos(x)$ and $g(x)=4$.
The line lies in between the range of the wave function so there will be two intersections for each period of the wave.
I want to find where on the $x$-axis those intersections will occur for a single period. Without resorting to numerical methods to approximate the roots and just using a plain scientific calculator, I'm trying to find both of them.
Setting both functions to equal each other yields me the first intersection, how do I get the next one in the same period?
$$\begin{array}{l}5\cos(x) = 4\\
\cos(x) = 0.8\\
x = \arccos(0.8)\\
x = 0.6435
\end{array}$$


Assume that there is an angle $\phi$ (in radians), such that $\cos \phi = 0.8$. Then, you can use the formula:
$$ \cos x = \cos \phi \implies x = 2k \pi \pm \phi, \; k \in \mathbb Z$$
This formula covers the phase shift case. For example:
$$\cos(x+a) = \cos \phi \implies x+a = 2k\pi \pm \phi,\; k \in \mathbb Z$$

We need to solve the equation:
$$5 \cos (x +0.5) = 4 \implies \cos (x + 0.5 ) = 0.8.$$ 
An angle $\phi$ that satisfies $\cos\phi  = 0.8$ is $\phi = 0.643501$. Using the above formula, we have:
$$\cos(x + 0.5) = \cos(0.643501)\implies x +0.5 = 2k\pi \pm 0.643501, \; k \in \mathbb Z .$$


*

*Now, for $k = 0$ and taking the plus sign, we have that $\boxed{x_1 = 0.643501- 0.5= 0.143501}$

*For $k = 1$ and taking the minus sign, we have that $\boxed{x_2 = 2\cdot \pi - 0.643501 -0.5 =5.13968}$
