To solve $\tan 316^\circ$, we first need to determine the reference angle $\alpha$, which is the acute angle formed with the x-axis.

Step 1: Find the reference angle

The angle $316^\circ$ is in the fourth quadrant (since it's between $270^\circ$ and $360^\circ$). To find the reference angle $\alpha$ in the fourth quadrant, we subtract the given angle from $360^\circ$:

$\alpha = 360^\circ - 316^\circ = 44^\circ$

So, the reference angle is $44^\circ$.

Step 2: Determine the sign

In the fourth quadrant, the tangent function is negative, so: $\tan 316^\circ = -\tan 44^\circ$

Therefore, the exact value of $\tan 316^\circ$ is $-\tan 44^\circ$.

Would you like further clarification on any of these steps?

Here are 5 questions to deepen understanding:

1. How do we find the reference angle for angles in different quadrants?
2. Why is the tangent function negative in the fourth quadrant?
3. What are the reference angles for $240^\circ$ and $150^\circ$?
4. How does the sign of the tangent change across the four quadrants?
5. Can you explain the general process for finding trigonometric values using reference angles?

Tip: Always remember that the tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants.