Anyone can help me on this qns?

The Fourier series expansion for the periodic function,f(t) = |sin t|is defined in its fundamental interval. Taking π = 3.142, calculate the Fourier cosine series approximation of f(t), up to the 6th harmonics when t = 1.09. Give your answer to 3 decimal places.

To calculate the Fourier cosine series approximation of the function f(t) = |sin t| up to the 6th harmonics when t = 1.09, we need to follow a series of steps. Here's how you can do it:

Step 1: Determine the fundamental period of the function f(t). In this case, the function f(t) = |sin t| is a periodic function with a period of 2π because the absolute value of sin t repeats every 2π.

Step 2: Write down the Fourier cosine series formula for f(t):

f(t) = a₀/2 + Σ(aₙ*cos(nωt))

In this formula, a₀/2 represents the DC component, aₙ represents the nth harmonic, ω represents the fundamental angular frequency, and t represents the point in time at which we want to calculate the approximation.

Step 3: Calculate the values of the coefficients a₀/2 and aₙ. Since f(t) is an even function (symmetric about the y-axis), the Fourier cosine series will only contain cosine terms. The coefficients a₀/2 and aₙ can be determined using the formulas:

a₀/2 = (2/π) * ∫(f(t) * cos(0ωt) dt) for the DC component (n = 0)
aₙ = (2/π) * ∫(f(t) * cos(nωt) dt) for the harmonic terms (n > 0)

Step 4: Substitute the value of t = 1.09 into the Fourier cosine series formula and calculate the approximation up to the 6th harmonics. Use the value of π as 3.142 for the calculations.

Given that t = 1.09, we can calculate the ω (angular frequency) as follows:
ω = 2π / T = 2π / 2π = 1

Now, use the formulas mentioned in Step 3 to calculate the coefficients:

a₀/2 = (2/π) * ∫(|sin t| * cos(0ωt) dt)
= (2/π) * ∫(|sin t| * cos(0) dt)
= (2/π) * ∫(|sin t| * 1 dt)
= (2/π) * ∫(|sin t| dt)
= (2/π) * ∫(sin(t) dt) [since sine is positive in the positive half-cycle]
= (2/π) * (-cos(t)) + C₁

Now, for aₙ:
aₙ = (2/π) * ∫(|sin t| * cos(nωt) dt)
= (2/π) * ∫(sin(t) * cos(nωt) dt) [again, considering sine is positive in the positive half-cycle]
= (2/π) * ∫(sin(t) * cos(n*t) dt)

You can calculate these integrals using integration techniques or software. Once you have the values for a₀/2 and aₙ, substitute them along with the value of t = 1.09 and calculate the Fourier cosine series approximation up to the 6th harmonics using the formula from Step 2:

f(t) = a₀/2 + Σ(aₙ*cos(nωt))

Remember to round the final answer to 3 decimal places.