Find the fourth roots of
− 1/2 + (square root)3/2 i
Write the roots in trigonometric form.
A -
w 1=cos(35°)+isin(35°)
w2 =cos(125°)+isin(125°)
w3 =cos(215°)+isin(215°)
w4 =cos(305°)+isin(305°)
B -
w1 =cos(40°)+isin(40°)
w2 =cos(130°)+isin(130°)
w3 =cos(220°)+isin(220°)
w4 =cos(310°)+isin(3105°)
C - w =cos(20°)+isin(20°) 1
w2 =cos(110°)+isin(110°)
w3 =cos(200°)+isin(200°)
w4 =cos(290°)+isin(290°)
D -
w1 =cos(30°)+isin(30°)
w2 =cos(120°)+isin(120°)
w3 =cos(210°)+isin(210°)
w4 =cos(300°)+isin(300°)
E -
w1 =cos(25°)+isin(25°)
w2 =cos(115°)+isin(115°)
w3 =cos(205°)+isin(205°)
w4 =cos(295°)+isin(295°)
I'm studying for my pre cal exam. Can you please help me with this question?
let z = ( -1/2 + √3/2 i)
= (cos 120° + sin 120 i)
z^(1/4) = cos (120/4) + sin(120/4) i
= cos 30 + sin30 i ------> my primary solution.
there will be 4 roots, and 360/4 = 90°
by adding multiples of 90 to 30° we can get the other 3
z^(1/3)
= cos 30 + sin 30 i
or
= cos 120 + sin 120 i
or
cos 210 + sin 210 i
or
cos 300 + sin 300 i
looks like D
looks like you are reviewing De Moivre's Theorem.
In general , in finding the nth root, there will be n such roots.
Once you have the primary one, divide 360 by n, then repeatedly add 360/n degrees to get the others.
Notice if you add one more at that point, it brings you back to your first solution.
e.g. adding 90° to cos 300 + sin300i will yield
cos 390 + sin 390 i
which is
cos 30 + sin 30 i by coterminal angles, 390 = 360 + 30
Sure, I'd be happy to help!
To find the fourth roots of the complex number −1/2 + √3/2i, we can express it in polar form and then use De Moivre's theorem.
In polar form, the complex number z = −1/2 + √3/2i can be written as r∠θ, where r is the magnitude of z and θ is the argument of z.
The magnitude of z is given by r = √((-1/2)^2 + (√3/2)^2) = 1.
The argument of z can be found using the inverse tangent function:
θ = arctan(√3/2 / (-1/2)) = arctan(−√3) = -π/3.
So, z = 1∠(-π/3).
Now, to find the fourth roots, we can use De Moivre's theorem, which states that for any complex number z = r∠θ,
z^1/n = (r^(1/n))∠(θ/n),
where n is the desired root.
For the fourth root (n = 4), we have:
z^1/4 = (1^(1/4))∠(−π/12).
Using the fact that 1^(1/4) = 1, we get:
z^1/4 = 1∠(−π/12).
Now, we can find the four roots by adding multiples of 2π/n to the argument:
w1 = 1∠(−π/12 + 0(2π/4)) = 1∠(−π/12) = cos(π/12) + isin(π/12),
w2 = 1∠(−π/12 + 1(2π/4)) = 1∠(5π/12) = cos(5π/12) + isin(5π/12),
w3 = 1∠(−π/12 + 2(2π/4)) = 1∠(9π/12) = cos(9π/12) + isin(9π/12),
w4 = 1∠(−π/12 + 3(2π/4)) = 1∠(13π/12) = cos(13π/12) + isin(13π/12).
Therefore, the correct answer is A -
w1 = cos(35°) + isin(35°),
w2 = cos(125°) + isin(125°),
w3 = cos(215°) + isin(215°),
w4 = cos(305°) + isin(305°).
I hope this helps! Let me know if you have any further questions.
To find the fourth roots of a complex number, we can use De Moivre's Theorem. De Moivre's Theorem states that for any complex number in trigonometric form, r(cosθ + isinθ), its nth roots can be found by taking the nth root of the magnitude r and dividing the angle θ by n.
In this case, let's find the fourth roots of the complex number -1/2 + √3/2i.
Step 1: Find the magnitude of the complex number.
The magnitude of a complex number is the square root of the sum of squares of its real and imaginary parts.
Magnitude = √((-1/2)^2 + (√3/2)^2) = √(1/4 + 3/4) = √4/4 = 1/2
Step 2: Find the angle of the complex number.
The angle can be found using the inverse tangent function.
Angle = atan(√3/2 / -1/2) = atan(-√3) = -π/3
Step 3: Find the fourth roots using De Moivre's Theorem.
To find the fourth roots, we divide the angle by 4 and take the fourth root of the magnitude.
w1 = (√(1/2))^1/4 * cos((-π/3) / 4) + i*sin((-π/3) / 4) = cos(-π/12) + i*sin(-π/12) = cos(35°) + i*sin(35°)
w2 = (√(1/2))^1/4 * cos((-π/3 + 2π) / 4) + i*sin((-π/3 + 2π) / 4) = cos(7π/12) + i*sin(7π/12) = cos(125°) + i*sin(125°)
w3 = (√(1/2))^1/4 * cos((-π/3 + 4π) / 4) + i*sin((-π/3 + 4π) / 4) = cos(11π/12) + i*sin(11π/12) = cos(215°) + i*sin(215°)
w4 = (√(1/2))^1/4 * cos((-π/3 + 6π) / 4) + i*sin((-π/3 + 6π) / 4) = cos(17π/12) + i*sin(17π/12) = cos(305°) + i*sin(305°)
So the fourth roots of -1/2 + √3/2i in trigonometric form are:
w1 = cos(35°) + i*sin(35°)
w2 = cos(125°) + i*sin(125°)
w3 = cos(215°) + i*sin(215°)
w4 = cos(305°) + i*sin(305°)
Therefore, option A is the correct answer.