Find the indicated roots. Write the answer in trigonometric form.
Fourth roots of 256(cos 220° + i sin 220°)
To find the fourth roots of a complex number in trigonometric form, we need to use De Moivre's Theorem.
De Moivre's Theorem states that for any complex number z = r(cosθ + i sinθ) and any positive integer n, the nth roots of z can be found by raising the magnitude, r, to the power of 1/n, and multiplying the angle, θ, by k/n, where k is an integer ranging from 0 to n-1.
In this case, we have the complex number 256(cos 220° + i sin 220°) and we want to find the fourth roots, so we need to find the value of (256)^(1/4) and multiply the angle by k/4.
First, let's find the magnitude of the complex number:
r = |z| = √[ (cosθ)^2 + (sinθ)^2 ]
= √[ (cos 220°)^2 + (sin 220°)^2 ]
= √[ cos^2(220°) + sin^2(220°) ]
= √[ 1 ]
= 1
Next, let's raise the magnitude to the power of 1/4:
r^(1/4) = 1^(1/4)
= 1
Finally, let's find the angle for each root by multiplying the original angle by k/4:
For k = 0: θ = (220° * 0/4) = 0°
For k = 1: θ = (220° * 1/4) = 55°
For k = 2: θ = (220° * 2/4) = 110°
For k = 3: θ = (220° * 3/4) = 165°
Putting it all together, the fourth roots of 256(cos 220° + i sin 220°) in trigonometric form are:
1(cos 0° + i sin 0°)
1(cos 55° + i sin 55°)
1(cos 110° + i sin 110°)
1(cos 165° + i sin 165°)