Question

4. (a) Express the following complex numbers in Cartesian form, (i.e. in the form z = x + iy with x, y ? ?) i. z = 2i (i + 2)² / (2 + 4i) ii. z = (1 / i) (e^8?i e^(?/2)i) / e^(1+?i) (b) Consider the following complex numbers: z? = 3i, z? = ?3 / 2 - i / 2 i. Express the complex numbers z? and z? in the polar form (z = re^i?) ii. Plot z?z?* as a vector in the complex plan (c) By expressing cos ? in terms of complex exponentials, express cos³ ? in terms of cos ? and cos 3?. (d) Consider the following equation z³ + 125 = 0 i. Calculate all the roots of this equation. ii. Represent all the roots as vectors in the complex plane.

          4. (a) Express the following complex numbers in Cartesian form, (i.e. in the form z = x + iy with x, y ? ?)
i. z = 2i (i + 2)² / (2 + 4i)
ii. z = (1 / i) (e^8?i e^(?/2)i) / e^(1+?i)
(b) Consider the following complex numbers:
z? = 3i, z? = ?3 / 2 - i / 2
i. Express the complex numbers z? and z? in the polar form (z = re^i?)
ii. Plot z?z?* as a vector in the complex plan
(c) By expressing cos ? in terms of complex exponentials, express cos³ ? in terms of cos ? and cos 3?.
(d) Consider the following equation z³ + 125 = 0
i. Calculate all the roots of this equation.
ii. Represent all the roots as vectors in the complex plane.

4. (a) Express the following complex numbers in Cartesian form, (i.e. in the form z = x + iy with x, y ? ?)
i. z = 2i (i + 2)² / (2 + 4i)
ii. z = (1 / i) (e^8?i e^(?/2)i) / e^(1+?i)
(b) Consider the following complex numbers:
z? = 3i, z? = ?3 / 2 - i / 2
i. Express the complex numbers z? and z? in the polar form (z = re^i?)
ii. Plot z?z?* as a vector in the complex plan
(c) By expressing cos ? in terms of complex exponentials, express cos³ ? in terms of cos ? and cos 3?.
(d) Consider the following equation z³ + 125 = 0
i. Calculate all the roots of this equation.
ii. Represent all the roots as vectors in the complex plane.

Added by Trinidad P.

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