Determine the two square roots of 3+j4 in both a)Cartesian form
b) Polar form
Any help with this question would appreciated
let the angle be Ø
then r = √(3^2+4^2) = 5
tanØ = 4/3, Ø = .9273
3 + 4j = 5(cos .9273 + jsin .9273)
by De Moivre's theorem
(3 + 4j)^(1/2) = √5(cos ((1/2).9273) + jsin ((1/2).9273))
= √5( cos .4636 + jsin .4636)
or 2 + j if expanded.
check:
if 2+j is the square root of 3+4j, then
(2+j)^2 should equal 3+4j
Left side = (2+j)^2
= 4 + 4j + j^2
= 4 + 4j - 1
= 3 + 4j
so √(3+4j) = 2+j or √5(cos .4636 + jsin .4636)
Sure! I can help you with that.
The given expression is √(3 + j4), where j represents the imaginary unit.
a) To determine the square roots of 3 + j4 in Cartesian form, we need to find two complex numbers in the form a + bj that satisfy (a + bj)^2 = 3 + j4.
Let's assume the square root is of the form x + yj. Expanding (x + yj)^2, we get:
(x + yj)^2 = x^2 + 2xyj + (yj)^2
= x^2 + 2xyj + y^2 * j^2
= x^2 + 2xyj - y^2
For the real parts to be equal, we have x^2 - y^2 = 3.
And for the imaginary parts to be equal, we have 2xy = 4.
From the first equation, we can rewrite x^2 = y^2 + 3. Plugging this into the second equation, we have:
2y(y^2 + 3) = 4
2y^3 + 6y - 4 = 0
We can solve this equation to find the value of y. Once we have y, we can substitute it back into the first equation to find the corresponding values of x.
b) To determine the square roots in polar form (r, θ), where r is the magnitude and θ is the angle, we need to find r and θ.
The magnitude r is given by the equation: r = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
The angle θ is given by the equation: tan(θ) = (Imaginary part) / (Real part) = 4 / 3. Taking the inverse tangent, we get: θ = tan^(-1)(4/3).
Thus, the two square roots of 3 + j4 are:
a) Cartesian Form:
x + yj (obtained after solving the equations)
For example, let's say the values of x and y are 1 and 2, respectively. Thus, one of the square roots in Cartesian form is 1 + 2j.
b) Polar Form:
(r, θ) = (5, tan^(-1)(4/3))
For example, one of the square roots in polar form is (5, tan^(-1)(4/3)).
Please note that the actual values of x, y, r, and θ will depend on the solutions of the equations above.
To determine the two square roots of 3 + j4 in both Cartesian and polar forms, we need to understand how to find the square root of a complex number.
Given a complex number in Cartesian form a + jb, where a and b are real numbers, we can find its square root by converting it to polar form using the modulus and argument.
1) Cartesian Form:
The given complex number is 3 + j4.
a) To find the square roots in Cartesian form, we let the square root be represented by z = x + jy. We square z and equate it to the given number:
(x + jy)^2 = 3 + j4
x^2 + 2xyj - y^2 = 3 + j4
By comparing the real and imaginary parts separately, we have two equations:
x^2 - y^2 = 3 (Equation 1)
2xy = 4 (Equation 2)
We can solve these simultaneous equations to find the values of x and y. Start by solving Equation 2 for y:
y = 2/x.
Substituting this value of y into Equation 1, we get:
x^2 - (2/x)^2 = 3
x^4 - 3x^2 - 4 = 0
This is a quadratic equation in terms of x^2. Solve this equation to find the values of x^2, and then determine the values of x. Once you have the value(s) of x, substitute them back into Equation 2 to find the corresponding value(s) of y.
b) The two square roots of 3 + j4 in Cartesian form will then be:
z1 = x1 + jy1
z2 = x2 + jy2
2) Polar Form:
To find the square roots in polar form, we need to first convert the complex number 3 + j4 into polar form, which consists of the modulus (r) and the argument (theta).
Given 3 + j4, we can find its polar form by calculating:
r = sqrt(a^2 + b^2)
theta = atan(b/a)
In this case, a = 3 and b = 4, so we have:
r = sqrt(3^2 + 4^2) = 5
theta = atan(4/3) ≈ 0.93 radians
The polar form of 3 + j4 is 5exp(j0.93) or simply 5∠0.93.
To find the square roots in polar form, we take the square root of the modulus (r) and divide the argument (theta) by 2.
The two square roots of 3 + j4 in polar form will be:
z1 = sqrt(r) * exp(j(theta/2))
z2 = sqrt(r) * exp(j(theta/2 + π))
Substituting the values, we get:
z1 = sqrt(5) * exp(j0.93/2)
z2 = sqrt(5) * exp(j(0.93/2 + π))