https://file.aiquickdraw.com/mathbot/file/17265860293610blvrth9.jpg

From the uploaded image, the task involves:

### Part 1: Matching Trigonometric Function Values
You are asked to label each trigonometric function with the corresponding value from the choices:
- (A) \(\frac{\sqrt{3}}{2}\)
- (B) \(\frac{1}{2}\)
- (C) \(\frac{\sqrt{2}}{2}\)

The trigonometric functions and their corresponding values are:
- **sin 30° = \(\frac{1}{2}\) (B)**
- **cos 30° = \(\frac{\sqrt{3}}{2}\) (A)**
- **cos 60° = \(\frac{1}{2}\) (B)**
- **sin 60° = \(\frac{\sqrt{3}}{2}\) (A)**
- **sin 45° = \(\frac{\sqrt{2}}{2}\) (C)**
- **cos 45° = \(\frac{\sqrt{2}}{2}\) (C)**

### Part 2: Using the Trigonometric Quotient Identity
The quotient identity is given as:

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

Using the values from Part 1, we can compute the following:

- \( \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)
- \( \tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \)
- \( \tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \)

Would you like any further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How can you use the unit circle to confirm the values of sine and cosine for standard angles?
2. What is the geometric interpretation of the tangent function on the unit circle?
3. How do you derive the values of sine and cosine for other angles like 0° and 90°?
4. Can you apply these trigonometric identities to solve real-world triangle problems?
5. How would you find the exact values for angles not commonly memorized, like 75° or 15°?

**Tip:** Memorizing the sine, cosine, and tangent values for 30°, 45°, and 60° will make solving trigonometry problems faster!

Evaluate the trigonometric functions and find the tangent of angles using the quotient identity.

Learn how to evaluate sine, cosine, and tangent functions for standard angles (30°, 45°, 60°) using the quotient identity. This problem helps to solidify key trigonometric concepts, including the sine, cosine, and tangent of standard angles. Suitable for students in Grades 9-11.

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