Periods of Trigonometric Functions

[mcfetridges]

Here are two general questions

How would you find the period of:

sin(2Pi*t)+sin(4Pi*t)

or

cos(3t)sin(2t)

Thanks

 

[rayjohn01]

The first is simple it is one period of the lower frequency ( the other is a simple harmonic ) . In the second did you really mean multiply or just leave out a + sign??
I do not always find the maths simple --- my fall back to this ( to get a clue ) is to graph the function. ( But not by hand ) .
I always use QBasic in which you can set up the equations and the graph in a matter of minutes .
However cos(a).cos(b) == a function of a+b and a-b so one frequency is 5 and the other 1 , so the frequency compared to either of the originals is 1.
That is, due to multiplication beats are formed between two frequencies
Since the normal wave equation is A.Sin ( 2.pi/T.t) it , means that 2.pi/T=1
so T = 2.pi
To solve these equations for T -- first compare them to the usual equation

The multiplier of t is 2.pi.f for a simple wave or 2.pi/T --- then if required use the normal trig relations for compound functions .
In cases of addition of sine waves there will only be a common period if the frequencies have an integer relation.
The same is true for multiplications . In general there may be no period at all, or maybe extremely long. For instance cos(99.t).cos(101.t) will will have a period about 50x times greater than either .
Ray.

 

[ravenprp]

for your question! To find the period of a trigonometric function, you need to look at the coefficient of the variable inside the parentheses. For example, in sin(2Pi*t)+sin(4Pi*t), the coefficient of t is 2Pi and 4Pi respectively. The period of a sine function is 2Pi, so the period of sin(2Pi*t) is 1 and the period of sin(4Pi*t) is 1/2. To find the period of the entire function, you need to find the least common multiple of these two periods, which is 1/2. Therefore, the period of sin(2Pi*t)+sin(4Pi*t) is 1/2.

For cos(3t)sin(2t), the period of cos(3t) is 2Pi/3 and the period of sin(2t) is Pi. The least common multiple of these two periods is 2Pi, so the period of the entire function is 2Pi.

I hope this helps! Let me know if you have any further questions.