Determining the period of \frac{\sin 2x}{\cos 3x}

Anne Wacker

Anne Wacker

Answered question


Determining the period of sin2xcos3x
I would like to compute the period of this function which is a fraction of two trigonometric functions.
Is there a theorem for this? what trick to use to easily find the period? I started by reducing the fraction but Im

Answer & Explanation

David Clayton

David Clayton

Beginner2021-12-29Added 36 answers

Suppose the period is p, and suppose the domain of the function is suitably defined, then for all values of x in the domain, we must have
If we set x=0, we have
sin(2p)=02pkπ, kZ
On the other hand, noting that sin2x=cos3x when x=π10, amongst other possible values, if we set x=π10, we get
3π10+3p=±(3π102p)+n2π, nZ
From this we get p=n2π or p=3π5+n2π
In order to satisfy this and the previous result for p we have to choose k and n such that both equations p=kπ2 and p=n2π are satisfied and p has minimum value, so we choose k=4 and n=1.
Therefore the period is 2π


Beginner2021-12-30Added 35 answers

If F(x) has a period T Then F(ax+b) has a period |(T/a)| If the expression is separated by + or - signs take the LCM of periods One of important concepts for determining period of function involving multiple periodic terms is that individual function should repeat simultaneously. This gives rises to LCM rule. It is defined for terms which appear as sum or difference in the function. However, LCM concept can be extended to division or multiplication of periodic terms as well.
Exceptions to LCM rule are important. Even function and function comprising of cofunctions are two notable exceptions to LCM rule. Besides, LCM of irrational periods of different kinds is not possible. This fact is used to determine periodic nature of function involving irrational periods. If LCM can not be determined, then given function is not periodic in the first place.


Expert2022-01-08Added 777 answers

sinx and cosx has a period of 2π. Therefore 2sin(x)4cos2(x)3 has a period of 2π
The fundamental period must be 2πn where nN. We have
Substituting x=(n2)πn we get
This is true only if n=1 or 2. If n=2 then
Therefore n2. So the fundamental period is 2π

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