Find the two square roots of 5-12i in the form a+bi, where a and b are real.
Mark on an Argand diagram the points P and Q representing the square roots. Find the complex number of R and S such that PQR and PQS are equilateral triangles.
How do you do the last part finding R and S?
This has been answered already.
I assume you can at least find P and Q.
Then you know that the angles involved are all 60 degrees, so just treat it like one of those navigation problems involving distances and bearings.
Using De Moivre's Theorem
let z = 5-12i
the r value is √(25+144) = 13
tanØ = -12/5
Ø = 292.619 ....°
so
z = 13cis292.619... (stored in calculators for accuracy), (cisØ is shortform for cosØ+i sinØ )
z^(1/2) = √13cis146.3099.. or √13cis(146.3099+180)
= -3 + 2i or 3 - 2i
√(5 - 12i) = 3-2i or -3+2i
On the Argand plane, mark 3-2i as P(3,-2) and -3+2i as Q(-3,2)
We can now treat the Argand plane just as if you had the standard x-y plane
PRQS will be a parallelogram and angles R and S are both 60°
if O is the "origin" in the Argand plane, the ROP will be a 30-60-90° right-angled triangle
slope OP = -2/3
slope of OR = 3/2
We know OP = √13, and RP = 2√13, so OR = √3√13 by comparison of ratios with the 30-60-90
from the slope of 3/2, we know tanØ = 3/2
Ø = appr 56.3099... (I stored this in my calculator)
OR = √3√13(cos 56.3099.. + i sin 56.3099..)
= 2√3 + 3√3 i ------> R = 2√3 + 3√3 i
similarly using symmetry, S = -2√3 - 3√3 i
I gave him a cut-and-paste copy of my previous solution.
Good question, like the way it comes out.
This is the 4th time this same person has posted this — and each under a different name (as if we couldn't figure it out!). =(
Here are the other 3:
https://www.jiskha.com/questions/1818615/find-the-two-square-roots-of-5-2i-in-the-form-a-bi-where-a-and-b-are-real-mark-on-an
https://www.jiskha.com/questions/1818622/find-the-two-square-roots-of-5-12i-in-the-form-a-bi-where-a-and-b-are-real-mark-on-an
https://www.jiskha.com/questions/1818617/find-the-two-square-roots-of-5-2i-in-the-form-a-bi-where-a-and-b-are-real-mark-on-an
To find the complex numbers R and S such that PQR and PQS are equilateral triangles, we first need to find the coordinates of P and Q on the Argand diagram.
Let's start by finding the square roots of 5-12i in the form a+bi.
Step 1: Express 5-12i in polar form.
The polar form of a complex number z = a+bi is given by z = r(cosθ + isinθ), where r is the magnitude of z and θ is the argument of z. The magnitude of z is given by r = √(a^2 + b^2), and the argument θ is given by θ = arctan(b/a).
For 5-12i, we have a = 5 and b = -12.
The magnitude of z is: r = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13.
The argument θ is: θ = arctan((-12)/5) = -1.176 radians (approximately).
Step 2: Find the principal square root of the magnitude.
Since we're looking for the square roots of 5-12i, we need to extract the square root of its magnitude. The principal square root of 13 is √13.
Step 3: Find the argument of the square roots.
The argument of the square roots will be half the argument of the original complex number. Therefore, we divide θ by 2: (-1.176)/2 = -0.588 radians (approximately).
Step 4: Convert the square roots to rectangular form.
Using the magnitude and argument calculated in the previous steps, we can express the square roots in rectangular form a+bi:
P = √13(cos(-0.588) + isin(-0.588))
Q = -√13(cos(-0.588) + isin(-0.588))
Now, let's move on to finding the complex numbers R and S required to create equilateral triangles PQR and PQS.
Step 5: Determine the angle for the equilateral triangle.
Since PQR and PQS are equilateral triangles, each angle of the triangle will be 60 degrees (or π/3 radians).
Step 6: Rotate P and Q to find R and S.
To find R and S, we need to rotate points P and Q by 60 degrees in the counterclockwise direction. This can be done by multiplying P and Q by the complex number e^(iπ/3), where e is the base of the natural logarithm.
R = P * e^(iπ/3) = √13(cos(-0.588 + π/3) + isin(-0.588 + π/3))
S = Q * e^(iπ/3) = -√13(cos(-0.588 + π/3) + isin(-0.588 + π/3))
These are the complex numbers R and S that correspond to the vertices of the equilateral triangles PQR and PQS.
To plot these points on the Argand diagram, you can use the rectangular form of the complex numbers (a+bi). Calculate the real part (a) and the imaginary part (b) separately for each complex number, and then mark the corresponding points on the diagram.