https://file.aiquickdraw.com/mathbot/file/1723884863261ki1ocjos.pdf

The last page of the document you uploaded is focused on revising Pythagoras and Trigonometry, particularly in 3D problems that involve two or more steps. Here’s an overview of the types of problems presented, specifically focusing on 3D trigonometry:

### **Problems Involving 3D Trigonometry:**

1. **Find EF**: 
   - This problem likely requires using Pythagoras' theorem in three dimensions, where \(EF\) could be a diagonal or a segment in a 3D shape.

2. **Find length PS**: 
   - This problem also involves finding the length of a segment \(PS\) within a 3D shape, possibly requiring the application of Pythagoras' theorem across different planes.

3. **Find angle PQS**:
   - This problem involves finding the angle between two segments or planes in a 3D shape, which might require using trigonometric ratios in conjunction with spatial reasoning.

4. **Find angle EDG**:
   - This problem asks for the calculation of another angle in 3D space, which again might involve breaking the problem into smaller 2D triangles within the 3D structure.

### **Steps to Solve 3D Trigonometry Problems:**

1. **Identify the 2D Planes**: Break down the 3D figure into relevant 2D triangles. This will often involve identifying the right-angle triangles within the 3D structure.

2. **Apply Pythagoras' Theorem**: Use Pythagoras’ theorem to find missing side lengths within the identified 2D triangles.

3. **Use Trigonometric Ratios**: To find angles, apply trigonometric ratios such as sine, cosine, and tangent to the appropriate triangles.

4. **Combine Results**: If the problem requires finding a length or angle across multiple planes or dimensions, combine your results using the Pythagorean theorem in 3D:  
   \[
   c = \sqrt{a^2 + b^2 + h^2}
   \]

5. **Recheck Your Work**: Ensure all steps account for the three-dimensional nature of the problem.

Would you like a detailed solution for one of these problems or more explanation on any specific concepts? 

### Follow-Up Questions:
1. Would you like a step-by-step solution to "Find EF" using Pythagoras' theorem in 3D?
2. Do you need help with a specific angle calculation, like finding angle PQS or EDG?
3. Are you interested in learning more about how to apply trigonometry in 3D shapes?
4. Would you like a recap on how to transition from 2D to 3D problems in trigonometry?
5. Do you have any questions about solving problems involving diagonals in 3D figures?

### Tip:
When dealing with 3D problems, sketching out the problem and labeling all known sides and angles can significantly simplify the process of applying the correct formulas.

Explore the last page of a document focusing on 3D trigonometry problems involving finding lengths like EF and PS, and angles like PQS and EDG. Learn step-by-step solutions and apply Pythagoras' theorem in 3D. Suitable for high school students studying trigonometry in three dimensions.

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