- #1
gongo88
- 3
- 0
Is there a way to calculate the period of a trigonometric series (like the one below) analytically?
x(t)=5sin(16t)-4cos(8t+3.1)
x(t)=5sin(16t)-4cos(8t+3.1)
Finding the period of trig series analytically?
In summary, the period of a trigonometric series can be calculated analytically by finding the least common multiple of the individual periods of each term. This can be done by setting the argument of each term equal to a multiple of 2π and finding the least common multiple of the resulting values. If the ratio of the individual periods is a rational number, the sum of the two functions will also be periodic with a period equal to the least common multiple. However, if the ratio is irrational, the period cannot be calculated analytically and must be determined graphically.
- #2
mathman
Science Advisor
- 8,140
- 572
The period in this case can be gotten from: 8t = 2π or t = π/4. This works since the other term has a period 16t = 2π or t = π/8 which is a harmonic.
- #3
- #4
- 9,568
- 775
The sum of two periodic functions will be periodic it the two periods are commensurable. That means the ratio of their periods is a rational number. And in that case, the period is the least common multiple of the individual periods. For non-integers p and q, the LCM is the least number z such that ap = z and bq = z for a and b integers.
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- #5
CellCoree
- 1,490
- 0
Yes, there is a way to calculate the period of a trigonometric series analytically. The period of a trigonometric function is defined as the smallest positive value of t for which the function repeats itself. In other words, it is the length of one complete cycle of the function.
To find the period of a trigonometric series, we first need to identify the coefficients of sine and cosine functions and their frequencies. In the given example, the coefficient of sine function is 5 and its frequency is 16, while the coefficient of cosine function is -4 and its frequency is 8.
The period of a trigonometric series can be calculated by taking the least common multiple (LCM) of the frequencies of the sine and cosine functions. In this case, the LCM of 16 and 8 is 16, so the period of the given series is 16 units of time.
To verify this, we can plot the function on a graph and see that the function repeats itself after every 16 units of time. This method can be applied to any trigonometric series to find its period analytically.
Related to Finding the period of trig series analytically?
1. What is a trigonometric series?
A trigonometric series is an infinite sum of terms involving trigonometric functions, such as sine and cosine. It can be written in the form of a_n * cos(nx) + b_n * sin(nx) where a_n and b_n are coefficients and n is a positive integer.
2. How do I find the period of a trigonometric series?
To find the period of a trigonometric series analytically, you can use the formula T = 2pi/n, where n is the coefficient of the highest power of x in the series. This means that the period is equal to 2pi divided by the coefficient of x in the series.
3. Can I use any trigonometric identity to find the period?
No, you cannot use any trigonometric identity to find the period. The only identity that can be used is the fact that cos(nx) and sin(nx) have a period of 2pi/n, as mentioned in the previous answer.
4. How do I know if a trigonometric series is periodic?
A trigonometric series is periodic if it repeats itself after a certain interval. This means that there exists a value T for which the function f(x) = a_n * cos(nx) + b_n * sin(nx) is equal to f(x + T) for all values of x. If this condition is satisfied, then the series is periodic.
5. Is it possible to find the period of a non-periodic trigonometric series?
No, it is not possible to find the period of a non-periodic trigonometric series. A non-periodic series does not repeat itself after a certain interval, so there is no single value T that could satisfy the condition mentioned in the previous answer. In this case, the series is considered aperiodic.
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