Question

Write the complex number in polar form. express the argument in degrees. 4i

a. 4(cos 0° + i sin 0°)
b. 4(cos 270° + i sin 270°)
c. 4(cos 90° + i sin 90°)
d. 4(cos 180° + i sin 180°)

Answer 1

The complex number 4i in polar form is 4(cos 90° + i sin 90°), which matches choice (c). The magnitude is 4, and the argument is 90° since 4i lies on the positive imaginary axis.

Step-by-step explanation:

To convert the complex number 4i to polar form and express the argument in degrees, we should consider the general polar form of a complex number which is r(cos θ + i sin θ), where r is the magnitude (modulus) and θ is the argument of the complex number. The complex number 4i has a real part of 0 and an imaginary part of 4. Therefore, its magnitude is 4 and it lies on the positive imaginary axis. In the complex plane, this corresponds to an angle of 90° or 270° from the positive real axis. Since it is on the positive imaginary axis, the correct angle is 90°.

The correct polar form of the complex number 4i is 4(cos 90° + i sin 90°), which corresponds to choice (c).

Answer 2

In analytical geometry, there are two types of roots: real roots and complex roots. Imaginary roots are those with the term 'i'. These are complex numbers that can be found on the number line, thus they are called imaginary. The term 'i' is equal to the expression √(-1). So when you solve these equations, just treat the complex numbers as is they are variables.

The only way to solve these is to test each choice such that its final answer would be 4i. These are important equivalents to know:

sin 0° = 0 cos 0° =1
sin 90° =1 cos 90° =0
sin 180° =0 cos 180° = -1
sin 270° = -1 cos 270° = 0

Using these values, let's simplify each choise.

A. 4(1 + i (0)) = 4
B. 4(0 + i (-1)) = -4i
C. 4(0 + i (1)) = 4i
D. 4((-1) + i (0)) = -4

The answer is C.