Question

Please answer all questions and explain every answer.

Lab Assignment 1: Complex Numbers 1. Find and plot all values of V-64. (hint: convert-64 to polar coordinates) 2. Prove that sin2z cos2z 1. hint: use the formulas for complex sin and cos) 3. Prove that cosh2z - sin2. (hint: use the formulas for complex sin and cos) 4. Use matlab to calculate each of the following quantities: cosh2π, tani, cos (ir) 5. Evaluate the following expressions and compare to the computer solution using matlab:ln(-e),2. (-1)stni 6. Let z ae t be the displacement of a particle from the origin at time t. Show that the particle travels in a circle of radius a at velocity v aw and with acceleration of magnitude U2/a directed towards the center of the circle. (hint: use the fact that u al ,a = Use Matlab to plot the roots of-8 (hint: there are three roots) d2z d dt2 with expressions for calculating magnitude of complex numbers.) 7.

Answer 1

4 squareroot - 67 z = (- 64)^1/4 z^4 = -64 convert - 64 into polar z = x + iy z = r e^i theta = r(cos theta + I sin theta) x = -64 y = 0 r = squareroot (- 64)^2 + 0^2 = 64 theta = tan^- 1 y/x theta = tan^- 1 theta = pi finding nth roots z = [r(cos(theta + 2 pi k) + i sin (theta + 2 pi k)]^1/4 = 64^1/4[cos(pi + 215 k/4) + I sin (pi + 2 pi k/4)] = 2 squaeroot 2 [cos(pi + 2 pi k/4) + I sin (pi + 2 pi k/4)] put k = 0, 1, 2, 3. K = 0 z_0 = 2 squareroot 2(cos pi/4 + i sin pi/4) = 2 squareroot 2(1/ squareroot 2 + i 1/ squareroot 2) z_0 = 2 + 2i k