Question

Evaluate each of the following complex numbers and express the result in rectangular form:

a. z1= 3ejπ/4
b. z3= 2e-jπ/2
c. z4=j3
d. z5= j-4

Answer 1

a.
`z_(1) = (3√(2))/(2)+j\cdot 3(√(2))/(2)`, b.
`z_(3) = -j\cdot 2`, c.
`z_(4) = -j`, d.
`z_(5) = 1`

Explanation:

All given complex numbers are in polar form, which are characterized by:

`z = r\cdot e^(j\cdot \theta)`

Where:

`r` - Magnitude, dimensionless.

`\theta` - Direction, measured in radians.

The rectangular form of complex numbers is represented by:

`z = r\cdot (\cos \theta + j\cdot \sin \theta)`

Now, each complex number is evaluated:

a.
`z_(1) = 3\cdot e^{j\cdot ( \pi)/(4) }`

`r = 3` and
`\theta = (\pi)/(4)`

`z_(1) = 3\cdot \left(\cos (\pi)/(4)+j\cdot \sin (\pi)/(4) \right)`

`z_(1) = (3√(2))/(2)+j\cdot 3(√(2))/(2)`

b.
`z_(3) = 2\cdot e^{-j\cdot (\pi)/(2) }`

`r = 2` and
`\theta = -(\pi)/(2)`

`z_(3) = 2\cdot \left[\cos \left(-(\pi)/(2)\right) + j\cdot \sin \left(-(\pi)/(2) \right) \right]`

`z_(3) = -j\cdot 2`

c.
`z_(4) = j^(3)`

By definition of complex number,
`j = √(-1)`, which means that
`j^(2) = -1`. Hence:

`z_(4) = j^(2+1)`

`z_(4) = j^(2)\cdot j`

`z_(4) = j\cdot j^(2)`

`z_(4) = -j`

d.
`z_(5) = j^(-4)`

By definition of complex number,
`j = √(-1)`, which means that
`j^(2) = -1`.

Hence:

`z_(5) = (1)/(j^(4))`

`z_(5) = (1)/(j^(2+2))`

`z_(5) = (1)/(j^(2)\cdot j^(2))`

`z_(5) = (1)/((-1)\cdot (-1))`

`z_(5) = 1`