To find the trigonometric function values for $319^\circ$, we can use the periodicity of trigonometric functions. First, note that:

• $319^\circ = 360^\circ - 41^\circ$, so $319^\circ$ is in the fourth quadrant.

In the fourth quadrant, the values of trigonometric functions behave as follows:

• Sine ($\sin$) is negative.
• Cosine ($\cos$) is positive.
• Tangent ($\tan$) is negative.

Since $319^\circ = 360^\circ - 41^\circ$, we use the following relationships:

• $\sin(319^\circ) = -\sin(41^\circ)$
• $\cos(319^\circ) = \cos(41^\circ)$
• $\tan(319^\circ) = -\tan(41^\circ)$

Given the values:

• $\sin(41^\circ) = 0.6561$
• $\cos(41^\circ) = 0.7547$
• $\tan(41^\circ) = 0.8693$

We can now calculate the trigonometric values for $319^\circ$:

1. $\sin(319^\circ) = -0.6561$
2. $\cos(319^\circ) = 0.7547$
3. $\tan(319^\circ) = -0.8693$

These are the values of the trigonometric functions for $319^\circ$.

Would you like further details on these functions or need additional clarification? Here are some related questions you might find interesting:

1. What are the properties of trigonometric functions in different quadrants?
2. How do we use reference angles to find trigonometric values?
3. What is the significance of negative and positive values for trigonometric functions?
4. Can we use a calculator to verify these values for any angle?
5. How do we derive trigonometric values using the unit circle?

Tip: Understanding the unit circle is essential for quickly finding trigonometric function values for any angle!