Given Z = (5+i)/(2+3i). To express this in the form x+yi, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2+3i is 2-3i.

Step 1: Express Z in the form x+yi. Given Z = (5+i)/(2+3i). To express this in the form x+yi, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2+3i is 2-3i. Z = (5+i)/(2+3i) × (2-3i)/(2-3i) Z = ((5+i)(2-3i))/((2+3i)(2-3i)) Expand the numerator: (5+i)(2-3i) = 5(2) + 5(-3i) + i(2) + i(-3i) = 10 - 15i + 2i - 3i^2 Since i^2 = -1: = 10 - 13i - 3(-1) = 10 - 13i + 3 = 13 - 13i Expand the denominator using the difference of squares formula (a+b)(a-b) = a^2-b^2: (2+3i)(2-3i) = 2^2 - (3i)^2 = 4 - 9i^2 Since i^2 = -1: = 4 - 9(-1) = 4 + 9 = 13 Now substitute the expanded numerator and denominator back into the expression for Z: Z = (13 - 13i)/(13) Z = (13)/(13) - (13i)/(13) Z = 1 - i a) The complex number Z in the form x+yi is 1-i. Step 2: Show that Z = sqrt(2)(()/(4) - i()/(4)). From part (a), we have Z = 1 - i. To express a complex number x+yi in polar form r( + i), we find the modulus r = sqrt(x^2+y^2) and the argument = ((y)/(x)) (adjusted for quadrant). For Z = 1 - i, we have x=1 and y=-1. Calculate the modulus r: r = sqrt(1^2 + (-1)^2) r = sqrt(1+1) r = sqrt(2) Calculate the argument : Since x=1 (positive) and y=-1 (negative), Z lies in the fourth quadrant. The reference angle is given by = |(y)/(x)| = |(-1)/(1)| = 1. So, = ()/(4). For a complex number in the fourth quadrant, = -. = -()/(4) Now substitute r and into the polar form r( + i): Z = sqrt(2)((-()/(4)) + i(-()/(4))) Using the trigonometric identities (-) = and (-) = -: Z = sqrt(2)(()/(4) - i()/(4)) b) We have shown that Z = sqrt(2)(()/(4) - i()/(4)). Step 3: By equating the form in (a) to that in (b), show that ()/(4) = (1)/(sqrt(2)). From part (a), Z = 1 - i. From part (b), Z = sqrt(2)(()/(4) - i()/(4)). Equate the two expressions for Z: 1 - i = sqrt(2)(()/(4) - i()/(4)) 1 - i = sqrt(2)()/(4) - isqrt(2)()/(4) For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal. Equating the real parts: 1 = sqrt(2)()/(4) Divide by sqrt(2): ()/(4) = (1)/(sqrt(2)) c) We have shown that ()/(4) = (1)/(sqrt(2)).