Alice had been standing on the ground (Point A) and observing a brightly colored object resembling a bird on the top of a tree at a distance of 4 meters from the tree. She decided to get a closer look by moving 2 meters closer to the tree (Point B). After moving closer, she realized that the object was not a bird but something that she could catch. Then, she decided to catch it by climbing the tree, which had a height of 6 meters from the ground.

Using the above scenario, please answer the following questions showing step by step calculations and stating the formulae.  Determine whether angle A is larger than angle B. Make a conclusion about the comparison of angles when observing an object from a distance versus close.

To determine whether angle A is larger than angle B, we need to compare the angles of elevation from Points A and B to the top of the tree.

**Scenario Overview:**

- **Point A**: Alice is 4 meters away from the tree.
- **Point B**: Alice moves 2 meters closer, so she is now 2 meters away from the tree.
- **Height of the tree**: 6 meters.

We previously calculated these angles as:

- \(\theta_A\) (from Point A): \(56.31^\circ\)
- \(\theta_B\) (from Point B): \(71.57^\circ\)

Let's go through the calculations to confirm these values and conclude which angle is larger.

### Angle Calculation from Point A

#### Using the tangent function:
\[
\tan(\theta_A) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{6}{4}
\]

#### Solving for \(\theta_A\):

\[
\theta_A = \tan^{-1}\left(\frac{6}{4}\right) = \tan^{-1}(1.5)
\]

Using a calculator:

\[
\theta_A \approx 56.31^\circ
\]

### Angle Calculation from Point B

#### Using the tangent function:
\[
\tan(\theta_B) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{6}{2}
\]

#### Solving for \(\theta_B\):

\[
\theta_B = \tan^{-1}\left(\frac{6}{2}\right) = \tan^{-1}(3)
\]

Using a calculator:

\[
\theta_B \approx 71.57^\circ
\]

### Conclusion: Comparison of Angles

- **Angle from Point A (\(\theta_A\)):** \(56.31^\circ\)
- **Angle from Point B (\(\theta_B\)):** \(71.57^\circ\)

Thus, \(\theta_B > \(\theta_A\).

Therefore, **angle B is larger than angle A**.

### General Conclusion about Angle Comparison

1. **Observation from a Distance vs. Close-Up:**

   - **From a distance (Point A):** The angle of elevation (\(\theta_A\)) is smaller because the object (top of the tree) appears less steep relative to the observer's line of sight.
   - **Closer to the object (Point B):** The angle of elevation (\(\theta_B\)) is larger because the object appears steeper relative to the observer's line of sight.

2. **Conclusion:**

   - As you move closer to an object, the angle of elevation increases. Therefore, when observing an object from a distance, the angle of

Explore the comparison of angles of elevation in trigonometry using a real-world scenario. Determine which angle is larger when observing an object from different distances. Includes step-by-step calculations and formulas for tangent function applications.

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