Evaluate in the domain of complex variables.



a) [3(cos⁡〖55°〗+j sin⁡〖55°〗 )]^2 b) (5.0/15°)^3
c) (52/42°)^(1/2) d) ∜4

To evaluate these expressions in the domain of complex variables, we can use the properties of complex numbers and the rules of trigonometry. Let's evaluate each expression step by step.

a) To evaluate [3(cos⁡〖55°〗+j sin⁡〖55°〗 )]^2:

Step 1: Rewrite the expression using Euler's formula.
[3(cos⁡〖55°〗+j sin⁡〖55°〗 )]^2 = [3e^(j55°)]^2

Step 2: Square the magnitude and double the argument.
[3e^(j55°)]^2 = 3^2e^(j2×55°) = 9e^(j110°)

Therefore, the value of [3(cos⁡〖55°〗+j sin⁡〖55°〗 )]^2 in the domain of complex variables is 9e^(j110°).

b) To evaluate (5.0/15°)^3:

Step 1: Rewrite the expression using Euler's formula.
(5.0/15°)^3 = (5.0e^(j0°))/(15°)^3

Step 2: Simplify the denominator.
(5.0e^(j0°))/(15°)^3 = (5.0/15^3)e^(j0°)

Step 3: Simplify the fraction.
(5.0/15^3)e^(j0°) = (5.0/3375)e^(j0°) = 5/675e^(j0°)

Therefore, the value of (5.0/15°)^3 in the domain of complex variables is 5/675e^(j0°).

c) To evaluate (52/42°)^(1/2):

Step 1: Rewrite the expression using Euler's formula.
(52/42°)^(1/2) = (52e^(j0°))/(42°)^(1/2)

Step 2: Simplify the square root.
(52e^(j0°))/(42°)^(1/2) = 52e^(j0°)/√42°

Step 3: Multiply the numerator and denominator by the conjugate of the denominator to get rid of roots.
52e^(j0°)/√42° = (52e^(j0°))/√42° * (√42°/√42°) = (52e^(j0°)√42°)/(√42°)^2

Step 4: Simplify the denominator.
(52e^(j0°)√42°)/(√42°)^2 = (52e^(j0°)√42°)/42° = (52√42°)e^(j0°)

Therefore, the value of (52/42°)^(1/2) in the domain of complex variables is (52√42°)e^(j0°).

d) To evaluate ∜4:

Step 1: Take the square root of 4.
∜4 = ∛(4)^(1/2)

Step 2: Simplify the square root.
∛(4)^(1/2) = ∛2

Therefore, the value of ∜4 in the domain of complex variables is ∛2.