Finding Principale period of $\cos$ function

Question

Find principle period of $3\cos (2x-3)$.

Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$.

Please someone can help me how to do this problem?

Thank you,

Rutvik Sutaria.

Answer 1

$$\cos(2x-3)=\cos(2x-3+2\pi)=\cos\big(2(x+\pi)-3\big).$$ Then, if $f(x)=3\cos(2x-3)$,

$$f(x+\pi)=3\cos\big(2(x+\pi)-3\big)=3\cos(2x-3+2\pi)=\cos(2x-3)=f(x)$$

Therefore the period of your function is $\pi$

Answer 2

Note that the graph of $y=3\cos(2x-3)$ is just the graph of $y=3\cos(2x)$ shifted horizontally. By how much does not matter for this problem, the important thing is that the period does not change.

So our period is the same as the period of $3\cos(2x)$. The graph of $y=3\cos(2x)$ is just the graph of $y=\cos(2x)$, scaled in the vertical direction. So the period is the same as the period of $\cos(2x)$.

For the period of $\cos(2x)$, note that as $x$ travels from $0$ to $\pi$, the number $2x$ travels from $0$ to $2\pi$, a full principal period of the cosine function. So $\cos(2x)$ has principal period $\pi$, and therefore so does our given function.