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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?

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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.

What is the representation of tan 316° using the reference angle?

Learn how to find the value of tan(316°) by determining the reference angle and applying the correct sign. This trigonometry problem covers key concepts like reference angles, tangent functions, and the signs of trigonometric functions in different quadrants. Suitable for students in Grades 9-12.

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