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Question Number 109955 by toa last updated on 26/Aug/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{minimal}\mathrm{period}\mathrm{of}\mathrm{cosx}+\cos 3\mathrm{x}$$

Answered by mathmax by abdo last updated on 26/Aug/20

$$2\pi$$

Commented by toa last updated on 26/Aug/20

I know 2pi is the period, but can you explain better how it is the minimal period?

Answered by mathmax by abdo last updated on 26/Aug/20

$$\mathrm{the}\mathrm{period}\mathrm{of}\mathrm{cosx}\mathrm{is}2\pi \mathrm{and}\mathrm{the}\mathrm{period}\mathrm{of}\cos \left(3\mathrm{x}\right)\mathrm{is}\frac{2\pi }{3}$$$$2\pi >\frac{2\pi }{3}\Rightarrow \mathrm{T}=2\pi$$$$\mathrm{other}\mathrm{ex}\mathrm{cosx}+\cos \left(2\mathrm{x}\right)+\cos \left(3\mathrm{x}\right)$$$$\mathrm{perid}\left(\mathrm{cosx}\right)=2\pi ,\mathrm{period}\left(\cos \left(2\mathrm{x}\right)\right)=\pi ,\mathrm{period}(\cos \left(3\mathrm{x}\right)=\frac{2\pi }{3}$$$$\max (2\pi ,\pi ,\frac{2\pi }{3})=2\pi \Rightarrow \mathrm{T}=2\pi$$

Answered by PRITHWISH SEN 2 last updated on 26/Aug/20

$$\mathrm{period}\mathrm{of}\cos \mathrm{x}=2\pi \mathrm{and}\cos 3\mathrm{x}=\frac{2\pi }{3}$$$$\mathrm{let}2\pi =3\mathrm{T}\mathrm{then}\frac{2\pi }{3}=\mathrm{T}$$$$\mathrm{now}\mathrm{period}\mathrm{of}\cos \mathrm{x}+\cos 3\mathrm{x}=\mathrm{lcm}(3\mathrm{T},\mathrm{T})=3\mathrm{T}=2\pi$$

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