+0  
 
0
33
1
avatar

Find the period of the function f(x) = sin(x/2) + sin(x/3).

 Jun 17, 2020
 #1
avatar+8321 
0

By sum-to-product formula:

 

\(\sin \dfrac x2 + \sin \dfrac x3 = 2 \sin \dfrac{5x}{12} \cos\dfrac{x}{12}\)

 

The function \(g(x) = \sin \dfrac{5x}{12}\) has period \(\dfrac{24\pi}5\), which means \(g\left(\dfrac{24n\pi}5 + x\right) = g(x), n\in \mathbb Z\)

 

The function \(h(x) = \cos \dfrac{x}{12} \) has period 24 pi, which means \(h(24n\pi + x) = h(x), n\in \mathbb Z\)

 

Therefore \(f(24\pi + x) = 2g(24\pi + x) h(24\pi + x) = 2g\left(\dfrac{24\pi}5 \cdot 5 + x\right)h(24\pi \cdot 1 + x) = 2g(x) h(x) = f(x)\)

 

Therefore the period is \(24\pi\).

 

No smaller T satisfies \(f(T + x) = f(x)\)

.
 Jun 18, 2020

21 Online Users

avatar
avatar
avatar
avatar