Answer:
Ф = π + 2nπ.
Explanation:
The function cos(x) is a periodic function, which means that in different intervals it touch the same value and keeps repeating, as you can see in this image:
Actually, the period of this function is 2π (360°), which means that cos(x)= cos(x + 2π) = cos(x + 2nπ) with n being any integer.
Now that we clarified the problem of multiple angles for the same value, we can take a look at how does a circle interact with the cosine function. For that, we will use something called polar coordinates, which means changing the x axis by Rcos(Ф) and the y axis by Rsin(Ф), but this time, we will use an unit circle, which means that R=1
The blue line represents cos(Ф) and the green one represents sin(Ф). As you can see, when Ф = 0, cos(Ф) = 1 .
As the period is 2π, half a period is π, which is half a circle. On the other side of th blue line, cos(Ф) = -1, but the other side of that line is half a circle, which is a movement of Ф=π.
Now, cos(0 + π) = -1. Then cos(π) = -1.
Yet, that is only one result of Ф. As we saw before, there exist multiple Ф that gives us the same value, and the value repits itself in a period of 2π any number of times you want.
So, cos(π + 2nπ) = -1
Then, Ф = π + 2nπ.