A periodic function f(t), with period 2π is defined as,f(t) = c for 0 < t < πf(t) = -c for -π < t < 0where c = 1.4, Taking π = 3.142, calculate the Fourier sine series approximation up to the 5th harmonics when t = 1.63. Give your answer to 3 decimal places.
To calculate the Fourier sine series approximation up to the 5th harmonics for the given function, we need to determine the coefficients of the sine functions for each harmonics.
The Fourier sine series for a periodic function f(t) with period 2π is given by:
f(t) = a1 sin(t) + a2 sin(2t) + a3 sin(3t) + ... + an sin(nt)
where ai is the coefficient of the sine function for the ith harmonic.
To find the coefficients, we can use the formula:
ai = (2/π) ∫[0, π] f(t) sin(it) dt
For the given function:
- For 0 < t < π, f(t) = c = 1.4
- For -π < t < 0, f(t) = -c = -1.4
Now let's calculate the coefficients up to the 5th harmonics:
For i = 1:
a1 = (2/π) ∫[0, π] f(t) sin(t) dt
= (2/π) ∫[0, π] 1.4 sin(t) dt
= (2/π) * 1.4 * (-cos(t)) |[0, π]
= (2/π) * 1.4 * (-cos(π) - (-cos(0)))
= (2/π) * 1.4 * (1 + 1)
= 2.8/π
For i = 2:
a2 = (2/π) ∫[0, π] f(t) sin(2t) dt
= (2/π) ∫[0, π] 1.4 sin(2t) dt
= (2/π) * 1.4 * (-cos(2t)/2) |[0, π]
= (2/π) * 1.4 * (-cos(2π)/2 - (-cos(0))/2)
= (2/π) * 1.4 * (1/2 + 1/2)
= 2.8/π
For i = 3:
a3 = (2/π) ∫[0, π] f(t) sin(3t) dt
= (2/π) ∫[0, π] 1.4 sin(3t) dt
= (2/π) * 1.4 * (-cos(3t)/3) |[0, π]
= (2/π) * 1.4 * (-cos(3π)/3 - (-cos(0))/3)
= (2/π) * 1.4 * (-1/3 + 1/3)
= 0
For i = 4:
a4 = (2/π) ∫[0, π] f(t) sin(4t) dt
= (2/π) ∫[0, π] 1.4 sin(4t) dt
= (2/π) * 1.4 * (-cos(4t)/4) |[0, π]
= (2/π) * 1.4 * (-cos(4π)/4 - (-cos(0))/4)
= (2/π) * 1.4 * (1/4 + 1/4)
= 0.7/π
For i = 5:
a5 = (2/π) ∫[0, π] f(t) sin(5t) dt
= (2/π) ∫[0, π] 1.4 sin(5t) dt
= (2/π) * 1.4 * (-cos(5t)/5) |[0, π]
= (2/π) * 1.4 * (-cos(5π)/5 - (-cos(0))/5)
= (2/π) * 1.4 * (-1/5 + 1/5)
= 0
Now, to calculate the Fourier sine series approximation for t = 1.63, we can substitute the coefficients into the series formula:
f(1.63) = (2.8/π) * sin(1.63) + (2.8/π) * sin(2 * 1.63) + (0) * sin(3 * 1.63) + (0.7/π) * sin(4 * 1.63) + (0) * sin(5 * 1.63)
Evaluating this expression gives the approximation for the given t value.