I want to find the period of the function f ( x ) = 2 cos

conIjonnoraj0nls

conIjonnoraj0nls

Answered question

2022-06-03

I want to find the period of the function
f ( x ) = 2 cos ( 3 x ) | sin ( 5 x ) | 3 | tan ( 8 x ) | cos 4 ( x ) 7 tan ( x 9 ) sec 3 ( 2 x 3 )
which is a rational function of trig functions.
So far I have found that:
1. the period of cos ( 3 x ) is 2 π 3 ;;
2. the period of | sin ( 5 x ) | is π 5 ;;
3. the period of cos 4 ( x ) is π;
4. the period of tan ( x / 9 ) is 9 π;
5. and the period of sec 3 ( 2 x / 3 ) is 3 π.
How do I find the total period?

Answer & Explanation

amerce5acih

amerce5acih

Beginner2022-06-04Added 1 answers

Each of the functions are 2 π k periodic, each for different k (since f ( x ) is a rational function of trig functions). You are missing a few, so I will fill in the full list:
1.For cos ( 3 x ), k = 1 / 3,
2.For | sin ( 5 x ) | , k = 1 / 10,
3. For | tan ( 8 x ) | , k = 1 / 16,
4. For cos 4 ( x ), k = 1 / 2,
5. For tan ( x / 9 ), k = 9 / 2,
6. For sec 3 ( 2 x / 3 ), k = 3 / 2.
Then the question is what is the least common multiple of all of these numbers (Somos above said gcd, I think you want l c m ). Why the l c m ? The least common multiple tells you what is the smallest number that is divisible by each of the above k. That is because each of the functions will are periodic with the least common multiple. We want the smallest since if f ( x ) is T periodic, it is also 2 T , 3 T , periodic. But we want the period T.
For the k values listed above, the l c m is 9. So the function is 18 π periodic. You can check this using your favorite software by checking | f ( x ) f ( x + 18 π ) | as long as you avoid the x-values for which the function blows up.

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