Find the sixth root of unity.

The sixth roots of unity are the solutions to the equation z^6 = 1. These roots are complex numbers that lie on the unit circle in the complex plane.

Step 1: Write down the equation for the sixth roots of unity: z^6 = 1.

Step 2: Convert the equation to polar form. In polar form, a complex number z is represented as z = r(cos θ + i sin θ), where r is the magnitude of z and θ is the angle that z makes with the positive real axis. Since we're looking for roots of unity, we know that r = 1. So our equation becomes cos(6θ) + i sin(6θ) = 1.

Step 3: Set the real and imaginary parts of the equation equal to each other. This gives us two equations: cos(6θ) = 1 and sin(6θ) = 0.

Step 4: Solve these equations for θ. The solutions to cos(6θ) = 1 are θ = 0, π/3, 2π/3, π, 4π/3, 5π/3. The solutions to sin(6θ) = 0 are the same.

Step 5: Divide each solution by 6 to get the sixth roots of unity. The sixth roots of unity are θ = 0, π/6, π/3, π/2, 2π/3, 5π/6.

Step 6: Convert these angles back to complex numbers using the polar form z = cos θ + i sin θ. The sixth roots of unity are 1, cos(π/6) + i sin(π/6), cos(π/3) + i sin(π/3), cos(π/2) + i sin(π/2), cos(2π/3) + i sin(2π/3), cos(5π/6) + i sin(5π/6).

These are the sixth roots of unity.

Sure, here is the step by step solution to find the sixth root of unity.

Step 1: The nth roots of unity are given by the formula: cos(2kπ/n) + isin(2kπ/n) where k = 0, 1, 2, ..., n-1.

Step 2: For the sixth root of unity, n = 6. So, we substitute n = 6 in the formula.

Step 3: The sixth roots of unity are: cos(2kπ/6) + isin(2kπ/6) where k = 0, 1, 2, 3, 4, 5.

Step 4: Calculate the roots one by one:

For k = 0, the root is cos(0) + isin(0) = 1.

For k = 1, the root is cos(π/3) + isin(π/3) = 1/2 + √3/2 i.

For k = 2, the root is cos(2π/3) + isin(2π/3) = -1/2 + √3/2 i.

For k = 3, the root is cos(π) + isin(π) = -1.

For k = 4, the root is cos(4π/3) + isin(4π/3) = -1/2 - √3/2 i.

For k = 5, the root is cos(5π/3) + isin(5π/3) = 1/2 - √3/2 i.

So, the sixth roots of unity are 1, 1/2 + √3/2 i, -1/2 + √3/2 i, -1, -1/2 - √3/2 i, and 1/2 - √3/2 i.

Sure, here are the steps to find the sixth roots of unity:

1. The nth roots of unity are given by the formula: cos(2πk/n) + isin(2πk/n) for k = 0, 1, ..., n-1.

2. For the sixth roots of unity, n = 6. So, we substitute n = 6 into the formula:

   cos(2πk/6) + isin(2πk/6) for k = 0, 1, ..., 5.

3. Now, we calculate the roots:

   • For k = 0: cos(0) + isin(0) = 1
   • For k = 1: cos(π/3) + isin(π/3) = 1/2 + i(√3/2)
   • For k = 2: cos(2π/3) + isin(2π/3) = -1/2 + i(√3/2)
   • For k = 3: cos(π) + isin(π) = -1
   • For k = 4: cos(4π/3) + isin(4π/3) = -1/2 - i(√3/2)
   • For k = 5: cos(5π/3) + isin(5π/3) = 1/2 - i(√3/2)

So, the sixth roots of unity are 1, 1/2 + i(√3/2), -1/2 + i(√3/2), -1, -1/2 - i(√3/2), and 1/2 - i(√3/2).