Question

What is the period of `f(t)=sin( t / 18 )+ cos( (t)/21 )`?

Answer 1

`252pi`

Explanation:

The periods dor both sin kt and cos kt is `2pi/k`

Here, the periods of the separate oscillations given by

`sin(t/18) and cos (t/21)` are `36pi and 42pi`, respectively,

For the compounded oscillation f(t), the period is given by

the period`P = 36 L pi = 42M pi`, for the least pair of positive

integers L and M. So, `P = 252 pi`, against L = 7 and M = 6.

See how it works.

`f(t+252pi)`

`=sin (t/18+14pi)+cos(t/21+12pi)`

`= sin(t/18)+cos(t/21)`

`=f(t)`.

Note that when this P is halved, the first term would change its sign..

Answer 2

`252pi`

Explanation:

Period of `sin (t/18)` --> `18(2pi) = 36pi`
Period of `cos (t/21)` --> `21(2pi) = 42pi`
Least common multiple of `36pi and 42pi`
`(36pi)` ... x (7) ---> `252pi`
`(42pi)`...x (6) ---> `252pi`
Period of f(t) --> `252pi`