Use Demoivre's Theorem to find the fourth roots of -16. Leave the four roots in trigonometric form. How the heck do i do this?
well, if you have 16@520; 16@180;16@900;16@1260
dividing by 4, roots are 4@130, 4@45;4@225;4@315
To use DeMoivre's Theorem to find the fourth roots of a number, -16 in this case, you need to follow these steps:
Step 1: Convert the number to its trigonometric form.
In this case, -16 can be represented as 16 * (-1), so we have -16 = 16 * cos(π) + i * sin(π).
Step 2: Apply DeMoivre's Theorem.
DeMoivre's Theorem states that for any complex number, z = r(cos θ + i sin θ), the nth roots of z can be found using the formula:
(sqrt(r) * cos(θ/n + (2kπ)/n) + i * sin(θ/n + (2kπ)/n))
where r is the magnitude or absolute value of z, θ is the argument or angle of z measured counterclockwise from the positive real axis, and k is an integer from 0 to n-1.
In this case, r = 16, and for -16, θ = π.
Step 3: Find the fourth roots.
To find the fourth roots, substitute the values for r, θ, and n into DeMoivre's Theorem. Since n = 4, we have:
sqrt(16) * cos(π/4), sqrt(16) * cos(π/4 + 2π/4),
sqrt(16) * cos(π/4 + 4π/4), sqrt(16) * cos(π/4 + 6π/4)
These simplify to:
4 * cos(π/4), 4 * cos(π/2), 4 * cos(3π/4), 4 * cos(π)
Now, calculating the cosine values:
4 * cos(π/4) ≈ 2.83 * (approximately)
4 * cos(π/2) = 0
4 * cos(3π/4) ≈ -2.83 * (approximately)
4 * cos(π) = -4
So, the four roots of -16 in trigonometric form are approximately 2.83, 0, -2.83, and -4.