Thank you for your very very clear explanations!You're on the right track. Basically you need to find the least common multiple of the periods of the two sinusoids.
Remember, the period of a signal is the reciprocal of the frequency. (T = 1/f)
After you calculate the periods of the two sinusoids (T = 1/f), you need to find the least common multiple of the two periods in order to get the period of the overall signal. Suppose, for example, you found that period1 = 4, and period2 = 6. Then, the least common multiple (LCM) of these two numbers (the smallest number that both numbers divide evenly into) is your answer. In this hypothetical case, we would have LCM(4 and 6) = 12. So, the overall period would be 12.
However, if period1 and/or period2 are fractions, then you should first multiply both periods by their least common denominator (so as to get rid of the denominators), and then find the LCM as was done above with the integers, and then divide your answer by the number that you multipled the two periods by in order to make them integers.
For example, suppose period1 = 2/3, and period2 = 4/7.
So, the LCM of the denominators (LCD) is LCM(3 and 7) = 21.
Remember, we doing this step in order to get rid of the denominators (make the fractions into integers).
So, 2/3 * 21 = 14, and 4/7 * 21 = 12. So now we find LCM(14 and 12) = 84. Now, divide the answer by 21 (the number we multiplied by above). So, 84/21 = 4. Thus, the overall period would be 4.
As another example, suppose period1 = 7, and period2 = 5/3.
So, we have 7*3 = 21
5/3*3 = 5
Now that we have converted our two numbers into integers by multiplying them both by 3, we have:
LCM(21 and 5) = 105.
So, we have period_overall = 105/3 = 35
Remember: in order to find the least common multiple of two numbers, you first write out all the prime factors of each number. Then, you choose from these two factor-lists, only the factors necessary so that the factors of each number are included.
For example, find LCM(25, 100)
25 = 5*5
100 = 5*5*2*2
LCM(25 and 100) = 5*5*2*2 = 100
Find LCM(24 and 14):
24 = 2*3*2*2
14 = 2*7
LCM(24 and 14) = 2*3*2*2*7 = 168 (notice that I included only three (not four) two's).
Hope that helps.
Oh, also, it doesn't really matter that one of the functions is a cosine and the other is a sine. Since cosine and sine only differ by phase, all the above will still work.
Also, be careful when graphing to pick a large enough window so as to see the the actual fundamental period of the signal. If you only look at a small section of the graph, it is easy to think that the fundamental period of the signal is much smaller than it actually is.
After you do it, tell us what answer you got.