Trouble Calculating for $30 \cos(30x)+14=−16$ I am unable to see how,
$$30\cos(30x)+14=−16$$
is equal to
$$\frac{\pi}{30} + n \frac{\pi}{15}$$
I solved up to this
$$\cos(30x) = -1$$
$$\pi= 30x$$
$$\frac{\pi}{30} = x$$
But I am unsure where the $\frac{\pi}{15}$ came from. Can someone help me understand this?
 A: Once you are at $$\cos(30x)=-1$$ your $180$ (which is degrees) is $\pi$ radians.  As the cosine function is periodic with period $2\pi$ we get 
$$30x=\pi+2k\pi$$
for any integer $k$.  Dividing by $30$ gives
$$x=\frac \pi{30}+\frac {k\pi}{15}$$
A: First of all, don't mix degrees and radians in the same solution without being explicit in what you're working with. It's bad practice, and simply incorrect if you're not very clear about what unit you're specifying.
So let's work only in radian measure.
$\displaystyle \cos(30x) = -1$
The reference angle with a cosine of $\displaystyle -1$ is $\displaystyle \pi$ radians. So $\displaystyle 30x = \pi$ is one solution.
However, there are an infinite number of solutions because of the periodicity of the cosine function, which has a period of $\displaystyle 2\pi$. So integer multiples of $\displaystyle 2\pi$ added to the reference angle will yield the same cosine.
So the full solution can be constructed as:
$\displaystyle 30x = \pi + 2n\pi, n \in \mathbb{Z}$
$\displaystyle x = \frac{\pi}{30} + \frac{n\pi}{15}, n \in \mathbb{Z}$
as required.
This is equivalent to the degree solution: $\displaystyle x = (6 + 12n)^{\circ}, n \in \mathbb{Z}$.
A: Substitute in $\pi/30$ and you get
$$\cos(30 \cdot \pi/30) = \cos \pi = -1.$$
Now repeat with some of your favorite integers $n$ and you'll see how it works. There is an infinite number of solutions to the equation: $\cos \pi, \cos 3\pi, \cos 5\pi, ... \cos -\pi, \cos -3\pi, ...$ all equal $-1$.
A: The thing is $\cos (x + 2n\pi ) = \cos x$ so if $\cos 30x = -1$ then $\cos 30(x+ \frac {2n\pi}{30})$ will also be equal to $-1$.
So when you got $\cos (30x ) = -1$ then 
$30x = \pi$ is ONE of the possible values for $30x$ (because $\cos \pi = -1$).
$30x = 3\pi$ is another one (because $\cos 3\pi = -1)$ and $30x = -pi$ and $30x=5\pi$ and $30x = 2147\pi$ are some more (because $\cos -\pi = \cos 5\pi = \cos 2147 \pi = -1$).
Indeed for any integer, $n$, you have an infinite number of possibilities of $30x = (2n+1)\pi$ becase $\cos (2n+1)\pi = -1$.
So $x =\frac {\pi}{30}$ is one answer and $\frac {\pi}{10}$ or $-\frac {\pi}{30}$ and $\frac {\pi}{6}$ and $\frac {2147\pi}{30}$ are all also solutions because for all thus values of $x$ you'd have $30x = \{\pi,3\pi, -pi,5\pi, 2147\pi\}$ and $\cos 30x = -1$ for all of those.
In fact, there are an infinite number of solutions.  For any $\frac{\pi}{30} + \frac n{15}\pi$ you will have $\cos 30x = \cos (30(\frac {\pi}{30} + \frac n{15}\pi))=\cos (\pi + 2n \pi) = \cos \pi = -1$.
So those are an infinite number of solutions where $\cos 30x = -1$
