The problem asks to find all values of $\theta$ in the interval $[0^\circ, 360^\circ)$ where the given condition holds:

$\text{tan}(\theta) \approx 0.8754064$

We will first solve for the principal angle using the inverse tangent function:

$\theta_1 = \tan^{-1}(0.8754064)$

Next, we will find the general solutions for $\theta$ because the tangent function repeats every $180^\circ$, and we are limited to the interval $[0^\circ, 360^\circ)$.

Let me compute the specific solutions for $\theta$.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the values using the following steps:

1. Compute $\theta_1 = \tan^{-1}(0.8754064)$ to get the principal solution.
2. Then, use the fact that the tangent function repeats every $180^\circ$ to find the second solution within the interval $[0^\circ, 360^\circ)$:
   • The second solution is $\theta_2 = \theta_1 + 180^\circ$.

If you calculate it, let me know if you'd like further help or explanations. Would you like to discuss any specific steps or concepts further?

Here are some related questions:

1. How do you calculate the arctangent of a number?
2. What is the significance of the principal angle in trigonometry?
3. Why does the tangent function repeat every 180 degrees?
4. How can you solve other trigonometric equations, such as for sine or cosine?
5. How do you handle solutions to trigonometric equations outside of the $[0^\circ, 360^\circ)$ interval?

Tip: When solving trigonometric equations, always check for all possible solutions within the specified interval by considering the periodicity of the function.