How should we find fundamental periods of
I)$$\cos(\frac{3x}{5})-\sin(\frac{2x}{7})$$
II)$$\frac{\sin(12x)}{1+\cos^2(6x)}$$
III)$$\sec^3(x)+\ {cosec}^3(x)$$
What i did was found the fundamental period of individual functions and took L.C.M and got correct answer. But then a thought occurred to me that their L.C.M is bound to be their period but it's not necessarily their fundamental period.
For example fundamental period of $|sin(x)|+|cos(x)|$ is $\frac{π}{2}$ ,so here fundamental period of $|\sin(x)|$ is ${π}$ and |{cos(x)}| is also $π$.
So how should i find these function's fundamental periods?How should i know where L.C.M is the fundamental period and when it is not?
Best Answer
Period of $f(x)=cos(3x/5)-\sin(2x/7)$ the periods of these two part are $10 \pi/3, 14\pi/$, the net period is LCM of these two numbers is $70\pi$
The period of $g(x)=\frac{\sin(12x)}{1+\cos^2(x)}$, is $\pi/6$ as $g(x+\pi/6)=g(x)$
The period of $h(x)= sec^3 x + cosec^3 x$ is $2 \pi$ as $h(x+2\pi)=h(x).$
Some Rules:
(1): If ratio of periods of $f_1$ and $f_2$ is rational ($T_1/T_2$=rational), then $F(x)=Af_1+Bf_2$ will be LCM of $T_1$ and $T_2$. Otherwise $F(x)$ is non-periodic. For example $\sin (x\sqrt{2})+\cos x$ is non periodic. $\sin (x \sqrt{2})+ \cos (x \sqrt{8)}$ is periodic.
(2): If $A=B$ and $f_1$ and $f_2$ interchange about a point $x=c$, then the period is $c$. LCM theorem does not give correct period. If $A \ne B$, depite intergange LCM Th. gives correct period. For example, $F(x)=\sin^4 x+ \cos^4 x$ has period $\pi/2$. But $G(x)=\sin^4 x -\cos^4 x$ has period $\pi$. $H(x)=4 \sin^4 x +7 \cos^4 x$ has period $\pi$.
(3):$$ LCM \left( \frac{a}{b}, \frac{c}{d}, \frac{e}{f} \right)= \frac{LCM(a,c,e)}{HCF(b,d,f)}$$