Find three cube roots for the following complex number. Leave your answers in trigonometric form. (Enter your answers from smallest to largest angle measure.)

To find the three cube roots of a complex number and express them in trigonometric form, we will follow these steps:

1. Break Down the Problem

We need to find the cube roots of a complex number, which can generally be represented as $z = r(\cos \theta + i \sin \theta)$ or $z = re^{i\theta}$. The formula for finding the $n$-th roots is given by: $z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right) \quad \text{for } k = 0, 1, \ldots, n-1$ Where:

• $r$ is the modulus of the complex number
• $\theta$ is the argument (angle)
• $n$ is the root we are calculating, in this case, $n = 3$

2. Relevant Concepts

Given a complex number $z$, the modulus $r$ can be calculated as: $r = |z| = \sqrt{x^2 + y^2}$ The argument $\theta$ (in radians) can be found using: $\theta = \tan^{-1} \left( \frac{y}{x} \right)$

3. Analysis and Detail

Assuming we have a specific complex number $z = x + iy$:

1. Calculate the modulus $r$.
2. Calculate the argument $\theta$.
3. Use the formula for cube roots given above.
4. Substitute $n = 3$ into the formula.

4. Verify and Summarize

Ensure calculations are correct, particularly checking angle measures.

Final Answer

The three cube roots of the complex number in trigonometric form will be calculated based on the specifics of the given complex number. Please provide the complex number $z$ to compute the cube roots.