Please solve
Write solutions on a separate sheet (s) of paper. Provide necessary derivations/explanations. Each exercise is worth 2 points. The assignment is due Thursday, April 21. For complex numbers z = 2 3i and w = -3 + 2i find (a) z + w, (b) z - w, (c) zw, (d) z/w Find the roots of x2 - 4x + 5 = 0 Find the roots of x2 + 2 alpha x + w2 = 0 (assume that w > alpha > 0) Using the idea on pp.3-4 of the tutorial, compute the square root of z = 5 + 12i and w = i. Give real and imaginary parts, complex conjugate and the modulus and argument of each of the complex numbers below: (a) z = -1, (b) z = 1 + i, (c) z = i, (d) -3 + 4i On the complex plane (Argand diagram), plot z = 2e2i pi/3 and w = 3e7i pi/12, zw, and z/w Use Euler's formula to prove De Moivre's formula: (cos x + i sin x)n = cos(nx) + i sin(nx) Provide derivations. Use De Moivre's formula to express a) cos(2x) and b) sin(3x) in terms of sin x: and cos x. Provide derivations. Use Euler's formula to prove the trig identities: a) cos(alpha + beta) = cos alpha cos beta -sin alpha sin beta, b) sin(alpha + beta) = sin alpha cos beta + cos alpha sin beta. Redo Exercise 4 using the polar form of complex numbers and Euler's formula. Express the answer in the cartesian (x+iy) form and compare with your previous answers.
Show transcribed image text Write solutions on a separate sheet (s) of paper. Provide necessary derivations/explanations. Each exercise is worth 2 points. The assignment is due Thursday, April 21. For complex numbers z = 2 3i and w = -3 + 2i find (a) z + w, (b) z - w, (c) zw, (d) z/w Find the roots of x2 - 4x + 5 = 0 Find the roots of x2 + 2 alpha x + w2 = 0 (assume that w > alpha > 0) Using the idea on pp.3-4 of the tutorial, compute the square root of z = 5 + 12i and w = i. Give real and imaginary parts, complex conjugate and the modulus and argument of each of the complex numbers below: (a) z = -1, (b) z = 1 + i, (c) z = i, (d) -3 + 4i On the complex plane (Argand diagram), plot z = 2e2i pi/3 and w = 3e7i pi/12, zw, and z/w Use Euler's formula to prove De Moivre's formula: (cos x + i sin x)n = cos(nx) + i sin(nx) Provide derivations. Use De Moivre's formula to express a) cos(2x) and b) sin(3x) in terms of sin x: and cos x. Provide derivations. Use Euler's formula to prove the trig identities: a) cos(alpha + beta) = cos alpha cos beta -sin alpha sin beta, b) sin(alpha + beta) = sin alpha cos beta + cos alpha sin beta. Redo Exercise 4 using the polar form of complex numbers and Euler's formula. Express the answer in the cartesian (x+iy) form and compare with your previous answers.
