Question

Write the complex number in rectangular form: 5л 511 2 cos +isin 6 6 3. Let 2,=5 cos 40° +i sin 40 °||2z=2 cos 100 °)+isin 100 |=10 cos (300° +i sin 300||Give the solutions to the following computations in polar form, with the angle in degrees between 0° and 360 a. 2,22=1 b. 2,23= c. (2, ²=i

Answer 1

Complex numbers can be converted from polar to rectangular form, and operations like multiplication or powers can be handled using De Moivre's Theorem by first performing the operation in polar form and then converting the result to rectangular form.

Step-by-step explanation:

To write the complex number in rectangular form, we convert the polar form to rectangular by using the formulas x = rcos(\u03b8) and y = rsin(\u03b8), where x is the real part and y is the imaginary part of the complex number. Given that z1 = 5 cos 40\u00b0 + i sin 40\u00b0, z2 = 2 cos 100\u00b0 + i sin 100\u00b0, and z3 = 10 cos 300\u00b0 + i sin 300\u00b0, calculating the products and powers of these complex numbers involves using De Moivre's Theorem.

For the computations:

• a. z1z2 = 1, means multiplying their magnitudes and adding their angles in polar form, then converting the result to rectangular form.
• b. z1z3, similar steps as in a, but with z1 and z3.
• c. (z1)\u00b2 = i, we square the magnitude of z1 and double its angle, then find its equivalence in rectangular form.