The period TT of a periodic function f(x)f(x) is given by
f(x)=f(x+T)f(x)=f(x+T)
Here,
f(t)=sin(t/30)+cos(t/42)f(t)=sin(t30)+cos(t42)
Therefore,
f(t+T)=sin(1/30(t+T))+cos(1/42(t+T))f(t+T)=sin(130(t+T))+cos(142(t+T))
=sin(t/30+T/30)+cos(t/42+T/42)=sin(t30+T30)+cos(t42+T42)
=sin(t/30)cos(T/30)+cos(t/30)sin(T/30)+cos(t/42)cos(T/42)-sin(t/42)sin(T/42)=sin(t30)cos(T30)+cos(t30)sin(T30)+cos(t42)cos(T42)−sin(t42)sin(T42)
Comparing,
f(t)=f(t+T)f(t)=f(t+T)
{(cos(T/30)=1),(sin(T/30)=0),(cos(T/42)=1),(sin(T/42)=0):}
<=>, {(T/30=2pi),(T/42=2pi):}
<=>, {(T=60pi),(T=84pi):}
The LCM of 60pi and 84pi is
=420pi
The period is T=420pi
graph{sin(x/30)+cos(x/42) [-83.8, 183.2, -67.6, 65.9]}
