sinusoidal equations I was doing some maths & encountered that  $$16\cos(8x) -2 = 6$$
and I looked for the answer and It's says " If we isolate the cosine function, we can use the inverse cosine function to find one value of 8x (the one between 0 degree and 180 degree)."
and my question why exactly 0° to 180°? couldn't that be between 0° to 360°?
 A: For any function $f$, the inverse $f^{-1}$ is only defined if $f$ is one-to-one. A one-to-one function $f$ is a function such that if $f(x)=f(y)$, then $x=y$. To put it another way, no two inputs to the function can have the same output.
The function $f(x)=\cos(x)$, where $0^{\circ}\leq x\leq 360^{\circ}$ is not one-to-one, as can be seen in the graph below:

The function $g(x)=\cos(x)$, where $0^{\circ} \leq x \leq 180^{\circ}$, is one-to-one, and so does have an inverse. This inverse is written as $\cos^{-1}$ or $\arccos$.
Functions always have a single output. It wouldn't make sense for $\cos^{-1}$ to output a number between $0^{\circ}$ and $360^{\circ}$ because then something like $\cos^{-1}(1/2)$ would be ambiguous: does it equal $60^{\circ}$ or $300^{\circ}$?
Instead, $\cos^{-1}(x)$ outputs a single number $\theta$ between $0^{\circ}$ and $180^{\circ}$ (this number is sometimes said to be the principal solution to the equation $\cos\theta = x$). Once you have found the principal solution, you can look for other ones.
A: As $\cos(t)=\cos(-t)$, a solution in $[0,180°]$ implies another in $[-180°,0°]$. Hence two per period, one per half-period.
A: cosine isn't a 1-1 function, there are an infinite number of answers for angles that have a cosine whose output is between $-1$ and $1$.  So the inverse cosine function, being a function has to give us exactly one answer.  We can get all of the answers in a single block if we just look for the one unique answer that is in the range of 0 to 180 degrees.  From there,  you can find the others by finding the angle in the other quadrant that shares the same cosine (quadrants 2 and 3,  quadrants 1 and 4),  and then using the fact that cosine is periodic with period $2\pi$ to add or subtract any multiple of that to your two answers to get every possible answer
