https://file.aiquickdraw.com/mathbot/file/1727120048820dy14kb4t.jpg

Let's analyze the matching problem in the image.

We are given a table with trigonometric function identities in different quadrants and a set of values to match with trigonometric functions at specific angles.

### Let's break down the matches:

1. **√3/2**
   - This value corresponds to the sine of a 60° reference angle. For the angle **120°** (in the second quadrant), the sine function remains positive.
   - So, **√3/2** matches with **sin(120°)**.

2. **-1/2**
   - This value corresponds to the cosine of a 60° reference angle. For the angle **120°** (in the second quadrant), the cosine function is negative.
   - So, **-1/2** matches with **cos(120°)**.

3. **√3/3**
   - This is a value associated with the tangent of a 30° reference angle. For the angle **120°**, the tangent is negative in the second quadrant, but the value does not match here.
   - Checking for **tan(210°)**, where the tangent is positive in the third quadrant, the value matches.
   - So, **√3/3** matches with **tan(210°)**.

4. **-√3/2**
   - This value corresponds to the cosine of a 60° reference angle. For the angle **210°** (in the third quadrant), the cosine function is negative.
   - So, **-√3/2** matches with **cos(210°)**.

5. **-√3**
   - This value corresponds to the tangent of a 60° reference angle. For the angle **120°** (in the second quadrant), the tangent function is negative.
   - So, **-√3** matches with **tan(120°)**.

### Final Matches:

1. **√3/2** → **sin(120°)**
2. **-1/2** → **cos(120°)**
3. **√3/3** → **tan(210°)**
4. **-√3/2** → **cos(210°)**
5. **-√3** → **tan(120°)**

Would you like a deeper explanation of how these trigonometric values are derived? Here are some related questions to expand on this:

1. How can you derive the sine and cosine values for angles in each quadrant?
2. What is the reference angle concept, and how is it applied to find trigonometric values?
3. How do the signs of trigonometric functions change across quadrants?
4. What are some methods to memorize the key trigonometric values (sin, cos, tan) for standard angles?
5. How can you use the unit circle to determine trigonometric function values for any angle?

**Tip**: When solving trigonometric problems involving quadrants, always check the signs of the functions based on the quadrant they are in!

Match the trigonometric function values to the corresponding angles using the provided table and options.

Solve a trigonometry matching problem using values for sine, cosine, and tangent across different quadrants. This problem covers key concepts like the unit circle, reference angles, and the behavior of trigonometric functions across quadrants. Suitable for students in Grades 10-12.

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