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Find the principal period of $$\sin\frac{3x}{4}+\cos\frac{2x}{5}$$

It was easy to find principle when single trigonometric function is given, but i don't know how to find principal period of sum of trigonometric function.

I have tried using graph. It didn't helped because it seems complicated to find the period using graph of this type of function.

My book has given solution, they have taken out period of each function and then assuming something m and n, seems too complicated because i think we can directly find the period, without assuming.

Gerry Myerson
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Freddy
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2 Answers2

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The period of $\displaystyle\sin(ax+c)=\frac{2\pi}a$ that of $\displaystyle\cos(bx+d)=\frac{2\pi}b$

Now if $\displaystyle\frac{\dfrac{2\pi}a}{\dfrac{2\pi}b}=\frac ba$ is rational, the period of $\displaystyle\sin(ax+c)+\cos(bx+d)$ will be lcm $\displaystyle\left(\frac{2\pi}b,\frac{2\pi}a\right)$ or its divisor

lab bhattacharjee
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    See also : http://mathcentral.uregina.ca/QQ/database/QQ.09.06/s/aakash1.html and http://math.stackexchange.com/questions/127154/what-is-the-periodicity-of-the-function-sinax-cosbx-where-a-and-b-ar – lab bhattacharjee Aug 15 '14 at 06:00
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Hint

Plotting the function is a good idea but your plot is not properly scaled. If you are not able to change the length of the $y$ axis, plot $$20\Big(\sin\frac{3x}{4}+\cos\frac{2x}{5}\Big)$$ and you will visually percieve what lab bhattacharjee means in his good answer.

Claude Leibovici
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