What is the period of f(t)=sin( t /6 )+ cos( (7t)/24 ) f(t)=sin(t6)+cos(7t24)?

2 Answers
Narad T.
May 12, 2018

The period is =12pi=12π

Explanation:

A periodic function f(x)f(x) is such that

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

where, TT is the period

Here,

f(t)=sin(t/6)+cos(7/24t)f(t)=sin(t6)+cos(724t)

f(t+T)=sin(1/6(t+T))+cos(7/24(t+T))f(t+T)=sin(16(t+T))+cos(724(t+T))

=sin(1/6t+1/6T)+cos(7/24t+7/24T)=sin(16t+16T)+cos(724t+724T)

Therefore,

f(t)=f(t+T)f(t)=f(t+T)

{(sin(t/6)=sin(1/6t+1/6T)),(cos(7/24t)=cos(7/24t+7/24T)):}

<=>, {(sin(t/6)=sin(t/6)cos(1/6T)+cos(t/6)sin(1/6T)),(cos(7/24t)=cos(7/24t)cos(7/24T)-sin(7/24t)sin(7/24T)):}

<=>, {(cos(1/6T)=1),(sin(1/6T)=0),(cos(7/24T)=1),(sin(7/24T)=0):}

<=>, {(1/6T=2pi),(7/24T=2pi):}

<=>, {(T=12pi),(T=48/7pi=48pi):}

The LCM of 12pi and 48pi is =12pi

The period is =12pi

graph{sin(x/6)+cos(7x/24) [-8.3, 56.63, -15.6, 16.88]}

Nghi N
May 13, 2018

48pi

Explanation:

Period of sin (t/6) --> 6(2pi) = 12pi
Period of cos ((7t)/24) --> (24(2pi))/7 = (48pi)/7
Period of f(t) is the least common multiple of 12pi and (48pi)/7.
12pi .....x (4) ........ --> 48pi
(48pi)/7 ...x (7) .... --> 48 pi

Period of f(t) --> 48pi

Related questions
See all questions in Amplitude, Period and Frequency
Impact of this question
2350 views around the world
You can reuse this answer
Creative Commons License