The goal of this exercise is to study euler’s important identity e ix = cos(x) + isin(x) and some of its consequences that will be useful in this course. (a) find the real and imaginary parts of the complex number e (2+ π 4 i) . (b) note that e ixe iy = e i(x+y) . use this fact and euler’s identity to rederive trigonometric formulas for sin(x+y) and cos(x+y) in terms of the quantities sin(x), sin(y), cos(x), cos(y). (c) for any a, b ∈ r, show that e (a+bi)x , e(a−bi)x are in the complex span of {e axcos(bx), eaxsin(bx)} by explicitly writing these two functions as complex linear combinations of e axcos(bx) and e axsin(bx). conversely, explain also how you can write each of the functions e axcos(bx), eaxsin(bx) as a complex linear combination of e (a+bi)x , e(a−bi)x . this shows that spanc{e (a+bi)x , e(a−bi)x } = spanc{e axcos(bx), eaxsin(bx)}. (d) consider the quantity bcos(µt)+csin(µt) where b, c, µ are positive real numbers. draw a right angle triangle with side lengths b, c and hypotenuse length a = √ b2 + c2 . let θ be the angle (in radians) between the side of length b and the hypotenuse. solve for b and c as functions of a and θ and use this to rewrite the quantity bcos(µt) + csin(µt) as a function of a, θ, µ, t, i. e., eliminate b and c. finally explain why bcos(µt) + csin(µt) = acos(µt − θ). a is called the amplitude of the final expression and θ is called the phaseshift. this final form of the expression is often called the phase-amplitude