EXERCISES: 1. Given the periodic signal x(t)=10cos²(2.5лt)+3cos(2πt)-8sin(3лt)cos(3лt)volts. (a) Determine manually the Fourier series of x(t) using trigonometric identities. Note: The Fourier series of x(t) is expressed as a sum of a constant, cosine terms, and sine terms. (b) Determine manually the period To of x(t) from its Fourier series as determined in (a). (c) Declare x(t) as a Scilab function and then plot it from -5

When a spring oscillates with decreasing amplitude (because it is being damped by friction, for example), its motion is often modeled as the product of a trigonometric function (the oscillation) and an exponential function (the decay of amplitude). Suppose that the displacement of a particular damped spring is s(t) 2e-1.8t cos(2πt), with s in inches and t in seconds. a) Find the spring's velocity with respect to t: v(t) = -4 pi ^(-1.8t) sin(2 pi ! t)-3.6e^(-1.8t) cos(2 pi t) That's not it. in/sec .-1.8t -4π¯¯ = -1.8t sin(2πt) — 3.6e¯ Preview v(t) = cos(2π. t) in/sec

Let W be the fraction of the width of the moon illuminated (as we observe it) as a function of time in days. For example, when the moon is full, W = 1. One model is W = + = 1/2 + 1/cos (255) 29.5 This model assumes that t = 0 corresponds to a full moon. This model for W represents a sine wave that has been shifted vertically. State the period of the sine wave. (Give an exact answer. Use symbolic notation and fractions where needed.) period: Explain in terms of phases of the moon what this period means. O This period represents the number of days between the phase of the new moon and the phase of the full moon. This period represents the number of days it takes the moon to get from any phase to that same phase again. O This period represents the number of days between the phase of the new moon and the phase of the half moon. Use the model for W to determine how much of the width of the moon is illuminated 2 days after a full moon. (Give your answer to the nearest whole number.) W = The…