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The image contains a trigonometry table where you are asked to construct a triangle to find missing values. The table lists various trigonometric functions (like sin, cos, sec, etc.) with associated angles in both degrees and radians. The goal is to fill in the missing function values.

I'll help you calculate these values based on the given angles and trigonometric functions.

Here is the breakdown for each function:

1. **sin(30°) or sin(π/6)**:
   - Function value: \( \frac{1}{2} \)

2. **cos(45°) or cos(π/4)**:
   - Function value: \( \frac{\sqrt{2}}{2} \)

3. **sec(π)**:
   - Function value: \( -1 \) 
   - Explanation: Secant is the reciprocal of cosine, and \( \cos(\pi) = -1 \), so \( \sec(\pi) = \frac{1}{\cos(\pi)} = -1 \).

4. **tan(30°) or tan(π/6)**:
   - Function value: \( \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \)

5. **cot(π/2)**:
   - Function value: \( 0 \)
   - Explanation: Cotangent is the reciprocal of tangent. Since \( \tan(\frac{\pi}{2}) \) is undefined (as it approaches infinity), \( \cot(\frac{\pi}{2}) = 0 \).

6. **csc(60°) or csc(π/3)**:
   - Function value: \( \frac{2}{\sqrt{3}} \) or \( \frac{2\sqrt{3}}{3} \)

7. **sin(π/3)**:
   - Function value: \( \frac{\sqrt{3}}{2} \)

8. **cos(60°) or cos(π/3)**:
   - Function value: \( \frac{1}{2} \)

9. **tan(π)**:
   - Function value: \( 0 \)
   - Explanation: \( \tan(\pi) = 0 \) because sine is 0 at π radians and cosine is -1.

10. **sec(45°) or sec(π/4)**:
    - Function value: \( \sqrt{2} \)

These values fill the missing entries in the table.

Would you like further explanations or clarifications on any of these calculations?

Here are five related questions to explore further:

1. What is the significance of constructing a right triangle to find trigonometric function values?
2. How does the unit circle relate to these trigonometric function values?
3. What are the reciprocal identities for trigonometric functions, and how are they used?
4. How would you calculate the value of trigonometric functions for angles greater than 90°?
5. What are the key differences between radian and degree measures in trigonometry?

**Tip:** Always remember that the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) can be a helpful tool when working with trigonometric functions.

Construct an appropriate triangle to find the missing values for various trigonometric functions like sin, cos, sec, tan, cot, csc, using angles in degrees and radians.

Learn how to find missing values of trigonometric functions such as sin, cos, sec, tan, cot, and csc using a triangle. This problem covers trigonometric identities and angle conversions for both degree and radian measures. Suitable for high school students in Grades 10-12.

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