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To solve this problem, we are asked to find approximate values for the given trigonometric expressions when \( \theta \) is small. In these cases, we can use small-angle approximations for sine, cosine, and tangent. These approximations are:

- \( \sin \theta \approx \theta \)
- \( \cos \theta \approx 1 - \frac{\theta^2}{2} \)
- \( \tan \theta \approx \theta \)

### Let's solve each part using these approximations:

#### Part (a): \( \frac{1 - \cos 2\theta}{\theta \sin \theta} \)

1. For small \( \theta \), we use the approximation for \( \cos 2\theta \):
   \[
   \cos 2\theta \approx 1 - 2\theta^2
   \]
   So,
   \[
   1 - \cos 2\theta \approx 1 - (1 - 2\theta^2) = 2\theta^2
   \]

2. Also, \( \sin \theta \approx \theta \). Therefore, the expression becomes:
   \[
   \frac{1 - \cos 2\theta}{\theta \sin \theta} \approx \frac{2\theta^2}{\theta \cdot \theta} = 2
   \]

Thus, for small \( \theta \), the expression in part (a) approximately equals 2.

#### Part (b): \( \frac{\cos 6\theta - 1}{\sin 3\theta \tan \theta} \)

1. For small \( \theta \), \( \cos 6\theta \approx 1 - 18\theta^2 \). Thus:
   \[
   \cos 6\theta - 1 \approx -18\theta^2
   \]

2. \( \sin 3\theta \approx 3\theta \) and \( \tan \theta \approx \theta \), so:
   \[
   \sin 3\theta \tan \theta \approx 3\theta^2
   \]

3. Substituting into the expression, we get:
   \[
   \frac{\cos 6\theta - 1}{\sin 3\theta \tan \theta} \approx \frac{-18\theta^2}{3\theta^2} = -6
   \]

Thus, for small \( \theta \), the expression in part (b) approximately equals \(-6\).

#### Part (c): \( \frac{\theta \tan \theta}{1 - \cos \theta} \)

1. \( \tan \theta \approx \theta \), so \( \theta \tan \theta \approx \theta^2 \).

2. Using \( \cos \theta \approx 1 - \frac{\theta^2}{2} \), we have:
   \[
   1 - \cos \theta \approx 1 - \left(1 - \frac{\theta^2}{2}\right) = \frac{\theta^2}{2}
   \]

3. Substituting, we get:
   \[
   \frac{\theta \tan \theta}{1 - \cos \theta} \approx \frac{\theta^2}{\frac{\theta^2}{2}} = 2
   \]

Thus, for small \( \theta \), the expression in part (c) approximately equals 2.

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### Summary of approximate values:
- (a) \( \frac{1 - \cos 2\theta}{\theta \sin \theta} \approx 2 \)
- (b) \( \frac{\cos 6\theta - 1}{\sin 3\theta \tan \theta} \approx -6 \)
- (c) \( \frac{\theta \tan \theta}{1 - \cos \theta} \approx 2 \)

Let me know if you need more details or clarifications!

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Here are 5 additional related questions for practice:

1. What is the small-angle approximation for \( \sin 2\theta \) and \( \tan 2\theta \)?
2. How does the approximation for \( \cos \theta \) change if higher-order terms are included?
3. Why is \( \theta \) assumed to be small in these approximations?
4. Can you derive the small-angle approximation for \( \sin(3\theta) \) using a series expansion?
5. How does the accuracy of small-angle approximations vary as \( \theta \) increases?

### Tip:
Small-angle approximations are extremely useful in physics, especially in mechanics and wave theory, where angles are often small enough to simplify trigonometric functions for easier computation.

When θ is small, find the approximate values of: a) (1 - cos 2θ) / (θ sin θ), b) (cos 6θ - 1) / (sin 3θ tan θ), c) (θ tan θ) / (1 - cos θ).

Find the approximate values of trigonometric expressions when θ is small, using small-angle approximations for sine, cosine, and tangent. Learn how to simplify expressions like (1 - cos 2θ) / (θ sin θ) and others with detailed explanations.

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