To find the six trigonometric (circular) functions for an angle of $390^\circ$, we can reduce it within the standard range by subtracting $360^\circ$:

$390^\circ - 360^\circ = 30^\circ$

Thus, the six trigonometric functions of $390^\circ$ are the same as those of $30^\circ$.

For $30^\circ$, the values of the six trigonometric functions are:

1. Sine: $\sin(390^\circ) = \sin(30^\circ) = \frac{1}{2}$

2. Cosine: $\cos(390^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2}$

3. Tangent: $\tan(390^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

4. Cosecant: $\csc(390^\circ) = \csc(30^\circ) = 2$

5. Secant: $\sec(390^\circ) = \sec(30^\circ) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

6. Cotangent: $\cot(390^\circ) = \cot(30^\circ) = \sqrt{3}$

Would you like more details on how each value was derived or have any questions?

Here are some additional related questions to explore:

1. How can trigonometric functions be derived for angles greater than $360^\circ$?
2. What is the significance of reference angles in trigonometry?
3. How do trigonometric functions relate to unit circle values?
4. Can trigonometric values of an angle be negative, and why?
5. What are some practical applications of trigonometric functions?

Tip: Remember that angles greater than $360^\circ$ are reduced by multiples of $360^\circ$ to find their equivalent angles within the first revolution of the unit circle.