Finding the fundamental period of $-6\cos(5\pi x)$

Question

I am trying to find the fundamental period of $ f(x) =-6\cos(5\pi x)$. I know a periodic function satisfies $f(x)=f(x+p)$.

I know that $\cos(x)$'s periodicity is $2\pi$ as $\cos(x+2\pi)=\cos(x)$ so I just assumed that $-6\cos(5\pi*1+2\pi) =-6\cos(5\pi)$, which it is but apparently this is not correct for the fundamental period?

If anyone can show me where I am getting confused and show me what is the correct way to do it I would be grateful!

Answer 1

Another way of thinking of the periodicity is this:

Let $f(x) = \cos(x)$. It is easy to show that $f(x) = f(x + 2\pi n)$ for $n \in \mathbb{N}$. In addition, $f'(x) = f'(x + 2 \pi n)$. Thinking about this problem whilst considering the derivative (rate of change) will show you that the period will involve an additional factor of $5$ i.e. $\cos(5\pi x)$ will oscillate more rapidly than $\cos(x)$, leading to a smaller period. As a note, the constant out front $(-6)$ will only change the scale of the function, not the period.

You can also consider $\cos(x)$ has a period of $2\pi$, meaning the length in $x$ required for the $\cos(x)$ waveform to complete one "iteration". With an argument of $5\pi x$, what value (length) of $x$ will make the overall argument $2\pi$?

$$5\pi x = 2\pi \implies x = \dfrac{2}{5}$$

Answer 2

The mistake you are making is that $$ f(x+p) = -6\cos\left(5\pi (x+p) \right) \neq -6\cos\left(5\pi x+p \right). $$

Answer 3

When someone writes $f(x) = f(x+p),$ they mean that for everything that you do with $x$ on the left side (for $f(x)$), you do the exact same thing to $x+p$ on the right side (for $f(x+p)$).

If $f(x) = -6\cos(5\pi x)$ then what happens to $x$ in $f(x)$ is that you multiply by $5\pi,$ then take the cosine of that result, then multiply that second result by $-6.$ So for $f(x+p)$ you have to multiply $x+p$ by $5\pi,$ then take the cosine, then multiply by $-6.$

One way to easily write this out is to take your expression for $f(x)$ (in this case it is $-6\cos(5\pi x)$), erase each occurrence of $x,$ and in each place where you erased $x$ insert $(x+p).$ The parentheses around $(x+p)$ are there to ensure that you're actually doing to $x+p$ what $f(x)$ did to $x,$ that is, multiplication by $5\pi$ should result in $5\pi(x+p)$ and not $5\pi x + p.$

Answer 4

Yes, cosine has period $2\pi$ so $\cos(5\pi x)$ will complete a period between $5\pi x= 0$, so $x= 0$, and $5\pi x= 2\pi$, so $x= \frac{2}{5}$.