# Angle of Elevation and Depression in Trigonometry

1. Introduction:

• The angle of elevation and depression is a concept in trigonometry that deals with the measurement of angles formed between a horizontal line and the line of sight to an object above or below the horizontal level. This concept is particularly useful in various real-world applications.

2. Angle of Elevation:

• Definition:

• The angle of elevation is the angle formed between the horizontal line of sight and the line of sight to an object above the horizontal.
• Example:

• When looking up at the top of a tree or a building, the angle formed between the horizontal line of sight and the line connecting the observer's eye to the top of the object is the angle of elevation.

3. Angle of Depression:

• Definition:

• The angle of depression is the angle formed between the horizontal line of sight and the line of sight to an object below the horizontal.
• Example:

• When looking down from the top of a hill or a tower, the angle formed between the horizontal line of sight and the line connecting the observer's eye to the bottom of the object is the angle of depression.

4. Measurement Units:

• Angles of elevation and depression are measured in degrees, minutes, and seconds, following the standard unit of angular measurement.

5. Trigonometric Functions:

• Tangent Function:
• The tangent of the angle of elevation ($\theta$) is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle formed by the horizontal line, the line of sight, and the line connecting the observer to the object:
• $\mathrm{tan}\left(\theta \right)=\frac{\text{Height}}{\text{Distance}}$

6. Problem-Solving with Angle of Elevation:

• Example Problem:
• A person is standing 50 meters away from the base of a tower. If the angle of elevation to the top of the tower is $4{5}^{\circ }$, find the height of the tower.
• Solution:
• Use the tangent function: $\mathrm{tan}\left(4{5}^{\circ }\right)=\frac{\text{Height}}{50}$ to find the height.

7. Problem-Solving with Angle of Depression:

• Example Problem:
• An observer on top of a lighthouse looks down at a ship at an angle of depression of $3{0}^{\circ }$. If the lighthouse is 100 meters high, find the distance between the observer and the ship.
• Solution:
• Use the tangent function: $\mathrm{tan}\left(3{0}^{\circ }\right)=\frac{100}{\text{Distance}}$ to find the distance.

8. Practical Applications:

• Surveying:
• In surveying, angles of elevation and depression are used to measure distances and elevations accurately.
• Astronomy:
• Astronomers use these angles to study celestial objects and calculate distances.
• Architecture:
• Architects use the concept for designing buildings and structures.

9. Summary:

• The angle of elevation and depression is a fundamental trigonometric concept used to measure angles formed between a horizontal line of sight and lines connecting observers to objects above or below the horizontal.