Evaluate(2+j4)*(-3-j4) and express in polar coordinate fform
I will assume your expression is a complex number of the form a + bi , let me know if you give it a different meaning.
(2+j4)*(-3-j4)
= -6 - 8j - 12j - 16j^2
= -6 - 20j + 16
= 10 - 20j
tanØ = -20/10 = -2, with Ø in quad IV
Ø = 5.176
argument = √(10^2 + (-20)^2) = 10√5
so we could write the result as
10√5cos 5.176 + 10√5sin 5.176
or
in short form: 10√5 cis 5.176
or, we could convert to polar first:
2+j4 = √20 cis 1.107
-3-j4 = 5 cis 4.068
multiply and you get Reiny's value.
To evaluate and express the product `(2 + j4) * (-3 - j4)` in polar coordinate form, we first multiply the complex numbers using the distributive property:
(2 + j4) * (-3 - j4)
= 2*(-3) + 2*(-j4) + j4*(-3) + j4*(-j4)
= -6 - 2j4 - 3j4 - 4
= -10 - 5j8
To express this complex number in polar coordinate form, we need to find its magnitude (r) and argument (θ).
1. Magnitude (r):
The magnitude (or modulus) of a complex number can be calculated using the Pythagorean theorem. Here, the real part (`-10`) is the negative of the hypotenuse, and the imaginary part (`-5j8`) forms the other two sides of a right triangle. Therefore, we can calculate the magnitude (r) as:
r = √((-10)^2 + (-5)^2)
= √(100 + 25)
= √125
= 5√5
2. Argument (θ):
The argument (or angle) of a complex number can be found using the inverse tangent function. We can calculate the argument (θ) as:
θ = atan(imaginary part / real part)
= atan((-5 * 8) / (-10))
= atan(40 / -10)
= atan(-4)
= -63.43° (rounded to two decimal places)
Therefore, the complex number (-10 - 5j8) in polar coordinate form is approximately `5√5 ∠ -63.43°`.