To determine the signs of the six trigonometric functions for the given angle of $-212^\circ$, let's break it down step by step:

Step 1: Understand the quadrant

• The angle $-212^\circ$ is a negative angle, meaning it is measured clockwise from the positive x-axis.
• To find its position, start from $0^\circ$ (the positive x-axis) and move clockwise by $212^\circ$.
• We know a full circle is $360^\circ$, so: $360^\circ - 212^\circ = 148^\circ$ This means the angle is in the third quadrant.

Step 2: Determine the signs of the functions in the third quadrant

In the third quadrant, the signs of the trigonometric functions are as follows:

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

Step 3: Assign the signs based on the quadrant

• Sine ($\sin(-212^\circ)$) is negative.
• Cosine ($\cos(-212^\circ)$) is negative.
• Tangent ($\tan(-212^\circ)$) is positive.

Final Answer:

• The trigonometric function value of $\sin(-212^\circ)$ is negative.
• The trigonometric function value of $\cos(-212^\circ)$ is negative.
• The trigonometric function value of $\tan(-212^\circ)$ is positive.

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Would you like more details on how to work with angles in different quadrants? Let me know if you have any questions!

Follow-up Questions:

1. How do you determine the exact values of sine, cosine, or tangent for an angle like $-212^\circ$?
2. How do you find the reference angle for a negative angle?
3. What are the other two quadrants and their function signs?
4. Can you explain why tangent is positive in the third quadrant?
5. How does a negative angle affect the trigonometric function values?

Tip: Remember that the signs of the trigonometric functions repeat every 90 degrees, so it’s always useful to determine the quadrant and use that to find the sign.