Simplify the following expressions by using trigonometric form and De Moivre's theorem
a. (√3-i)^4
b. (-4+4i√3)^5
c.(2+3i)^4
d. (4-i)^6
Tell you what -- why don't you simplify them and we'll check your answers.
To get started on (a)
√3-i = 2cis -π/6
so, (√3-i)^4 = 16cis -2π/3
now just convert that back to rectangular values.
but i don't get the first one
You need to review the methods of expressing complex numbers as a+bi and r cisθ. Surely your text has a discussion of the topic.
Also, a simple google on "complex numbers polar" will turn up may explanations and examples.
√3-i can be plotted as the point (√3,-1) in the x-y plane.
Now, measuring from the x-axis, that point is at an angle of -π/6, and is 2 units from the origin. Hence the polar form.
What do I do next after that?
To simplify these expressions using trigonometric form and De Moivre's theorem, we need to represent the complex numbers in their polar form and apply the theorem accordingly. Let's go through each expression:
a. (√3-i)^4:
Step 1: Find the magnitude and argument of the complex number (√3-i).
The magnitude (r) can be calculated as √((√3)^2 + (-1)^2) = √(3 + 1) = √4 = 2.
The argument (θ) can be calculated as tan^(-1)(-1/√3) = -π/6 (since -1/√3 is negative, the angle is in the 4th quadrant).
Step 2: Apply De Moivre's theorem with the polar form (r∠θ)^n = r^n∠(nθ):
(√3-i)^4 = (2∠(-π/6))^4 = 2^4∠(-4π/6) = 16∠(-2π/3).
b. (-4+4i√3)^5:
Step 1: Find the magnitude and argument of the complex number (-4+4i√3).
The magnitude (r) can be calculated as √((-4)^2 + (4√3)^2) = √(16 + 48) = √64 = 8.
The argument (θ) can be calculated as tan^(-1)(4√3/-4) = -π/3 (since 4√3/-4 simplifies to -√3 and is negative, the angle is in the 3rd quadrant).
Step 2: Apply De Moivre's theorem with the polar form (r∠θ)^n = r^n∠(nθ):
(-4+4i√3)^5 = (8∠(-π/3))^5 = 8^5∠(-5π/3) = 32768∠(-5π/3).
c. (2+3i)^4:
Step 1: Find the magnitude and argument of the complex number (2+3i).
The magnitude (r) can be calculated as √(2^2 + 3^2) = √(4 + 9) = √13.
The argument (θ) can be calculated as tan^(-1)(3/2) = π/3 (since 3/2 is positive, the angle is in the 1st quadrant).
Step 2: Apply De Moivre's theorem with the polar form (r∠θ)^n = r^n∠(nθ):
(2+3i)^4 = (√13∠(π/3))^4 = (√13)^4∠(4π/3) = 169∠(4π/3).
d. (4-i)^6:
Step 1: Find the magnitude and argument of the complex number (4-i).
The magnitude (r) can be calculated as √(4^2 + (-1)^2) = √(16 + 1) = √17.
The argument (θ) can be calculated as tan^(-1)(-1/4) = -π/6 (since -1/4 is negative, the angle is in the 4th quadrant).
Step 2: Apply De Moivre's theorem with the polar form (r∠θ)^n = r^n∠(nθ):
(4-i)^6 = (√17∠(-π/6))^6 = (√17)^6∠(-6π/6) = 24137569∠(-π).
Therefore, the simplified expressions using trigonometric form and De Moivre's theorem are:
a. (√3-i)^4 = 16∠(-2π/3)
b. (-4+4i√3)^5 = 32768∠(-5π/3)
c. (2+3i)^4 = 169∠(4π/3)
d. (4-i)^6 = 24137569∠(-π)