Doing transformations on trignometric functions

Question

I have a function $$f(x)=\sqrt{1-\cos(x)}$$ with the fundamental period $2\pi$. But I can also write this as $$\sqrt{2} \sin(x/2)$$ whose fundamental period is $4\pi$. Why has the fundamental period changed.

Answer 1

First of all, period of $\sqrt{2}\sin(x/2)$ is $4\pi$ not $\pi$. (Just check by putting some values, like is $\sqrt2\sin(0/2)=\sqrt2\sin(\pi/2)$?)

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The actual value of $\sqrt{1-\cos(x)}$ is $\sqrt2|\sin(x/2)|$ which indeed has a period of $2π$.

Hope this helps. :)

Answer 2

The two functions are not identical.

Take, for example, $x=-\pi$.

Then, $f(x) = \sqrt{1-\cos (-\pi)} = \sqrt{1 - (-1)} = \sqrt{2}$, while

$$\sqrt 2 \sin\left(-\frac \pi2\right) = -\sqrt{2}\cdot\sin\frac\pi2 = -\sqrt 2$$