Question

What is the period of `f(t)=sin( t /2 )+ cos( (7t)/24 )`?

Answer 1

The period is `=48pi`

Explanation:

A periodic function is such that

`f(x)=f(x+T)`

where `T` is the period

Therefore,

`sin(7/24t)=sin(7/24(t+T))`

`=sin(7/24t+7/24T)`

`=sin(7/24t)cos(7/24T)+sin(7/24T)cos(7/24t)`

Comparing the `LHS` and the `RHS`

`{(cos(7/24T)=1),(sin(7/24T)cos(7/24t)=0):}`

`<=>`, `{(7/24T=14pi):}`

`T=48pi`

`sin(t/2)=sin(1/2(t+T))`

`=sin(1/2t+1/2T)`

`=sin(t/2)cos(T/2)+sin(T/2)cos(t/2)`

Comparing the `LHS` and the `RHS`

`{(cos(T/2)=1),(sin(T/2)cos(t/2)=0):}`

`<=>`, `{(T/2=2pi),(T/2=0):}`

`T=4pi`

The `LCM` of `4pi "and" 48pi` is `=48pi`