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### Exercise 7.3 Questions:

1. **Question 1:**  
   Write the closest quadrant angles between which the angle lies.

   - \(\frac{7\pi}{6}\)
   - \(\frac{23\pi}{7}\)
   - \(5\pi\)
   - \(6.4\)
   - \(6.5\)
   - \(-1\)

2. **Question 2:**  
   Find the quadrant of \(\theta\) and values of its trigonometric ratios given:
   
   - \(\sin \theta > 0\), \(\cos \theta < 0\)
   - \(\cos \theta < 0\), \(\tan \theta > 0\)
   - \(\sin \theta < 0\), \(\cos \theta < 0\)
   - \(\sin \theta > 0\), \(\sec \theta < 0\)

3. **Question 3:**  
   Find the values of the remaining trigonometric ratios:
   
   - \(\cos \theta = -\frac{3}{5}\) and \(\theta\) lies in II quadrant.
   - \(\sin \theta = \frac{4}{5}\) and the terminal side is in IV quadrant.
   - \(\tan \theta = -\frac{1}{2}\) and the terminal side is not in I quadrant.
   - \(\sec \theta = 2\) and \(\theta\) lies in IV quadrant.
   - \(\csc \theta = -\frac{5}{3}\) and the terminal side is in I quadrant.

4. **Question 4:**  
   If \(\tan \theta = \frac{1}{\sqrt{3}}\) and \(\theta\) is in I quadrant, then prove that \(7\sec^2 \theta + 3\cot^2 \theta = 46\).

5. **Question 5:**  
   If \(\sin \theta = \frac{3}{5}\) and \(\theta\) is in I quadrant, find the value of \((\tan \theta + \sec \theta)\).

6. **Question 6:**  
   Find values of the trigonometric ratios without using a table or calculator:
   
   - \(420^\circ\)
   - \(5\pi\)
   - \(765^\circ\)
   - \(-675^\circ\)
   - \(7\pi\)

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### Relative Questions:
1. How do you determine which quadrant an angle lies in based on its value?
2. What is the significance of the signs of trigonometric functions in different quadrants?
3. How do you find the other trigonometric functions when one is given, particularly in a specified quadrant?
4. Can you explain how to prove identities involving trigonometric functions like in Question 4?
5. What methods can you use to find trigonometric values of angles without using a calculator?

### Tip:
Understanding the unit circle is key to solving trigonometric problems, as it helps in identifying the values of trigonometric functions in different quadrants.

The exercise includes questions related to trigonometric identities, functions, and angles. Some of the key problems involve identifying quadrants for given angles, determining trigonometric ratios in different quadrants, proving trigonometric identities, and finding specific trigonometric values without a calculator.

Explore detailed solutions to trigonometric problems focusing on quadrants, trigonometric identities, and ratio determination. Perfect for students in Grades 10-12, these exercises cover essential trigonometric concepts and provide insights into solving without a calculator.

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