Find the square roots of z=5−12i
z = 5 - 12i
|z| = √(25 + 144) = √169 = 13
tanθ = -12/5
θ = 292.62°
then 5-12i = 13cis 292.62°
(5-12i)^(1/2) = √13cis146.309..°
= √13cos146.309.. + √13sin146.309..
= -3 + 2i <---- primary root
second root = √13cos(146.309..+180) + √13sin(146.309..+180)
= 3 - 2i
To find the square roots of a complex number, such as z = 5 - 12i, we can use the following steps:
Step 1: Write the complex number in the form a + bi, where a is the real part and b is the imaginary part.
For z = 5 - 12i, we already have it in this form.
Step 2: Convert the complex number to polar form by using the magnitude and argument.
The magnitude (r) of a complex number z = a + bi is given by r = sqrt(a^2 + b^2).
For z = 5 - 12i, the magnitude (r) equals sqrt(5^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13.
The argument (θ) of a complex number z = a + bi is given by arctan(b / a) - in the range of (-π, π].
For z = 5 - 12i, the argument (θ) equals arctan((-12) / 5) = arctan(-2.4) ≈ -1.1903.
So, z in polar form is r(cos(θ) + isin(θ)) = 13(cos(-1.1903) + isin(-1.1903)).
Step 3: Obtain the square roots in polar form.
To find the square roots, we take the square root of the magnitude and halve the argument.
The square roots of z in polar form are sqrt(r)(cos(θ/2) + isin(θ/2)) and sqrt(r)(cos(θ/2 + π) + isin(θ/2 + π)).
For z = 5 - 12i, the square roots in polar form are:
sqrt(13)(cos(-1.1903/2) + isin(-1.1903/2)) ≈ sqrt(13)(0.826 + i(-0.325))
sqrt(13)(cos(-1.1903/2 + π) + isin(-1.1903/2 + π)) ≈ sqrt(13)(-0.826 + i(0.325))
Step 4: Convert the square roots back to rectangular form.
To convert a complex number from polar form to rectangular form, we use the formulas x = r * cos(θ) and y = r * sin(θ).
For the first square root:
x = sqrt(13) * cos(-1.1903/2) ≈ sqrt(13) * 0.826 ≈ 3.538
y = sqrt(13) * sin(-1.1903/2) ≈ sqrt(13) * -0.325 ≈ -1.435
So, the first square root of z = 5 - 12i is approximately 3.538 - 1.435i.
For the second square root:
x = sqrt(13) * cos(-1.1903/2 + π) ≈ sqrt(13) * -0.826 ≈ -3.538
y = sqrt(13) * sin(-1.1903/2 + π) ≈ sqrt(13) * 0.325 ≈ 1.435
So, the second square root of z = 5 - 12i is approximately -3.538 + 1.435i.
Therefore, the square roots of z = 5 - 12i are approximately 3.538 - 1.435i and -3.538 + 1.435i.