WHAT IS THE RELATIONSHIP BETWEEN THE ANGLE OF ELEVATION AND THE ANGLE OF DEPRESSION? GIVE AN EXAMPLE OF AN APPLICATION USING THE ANGLE OF ELEVATION OR THE ANGLE OF DEPRESSION
The Correct Answer and Explanation is:
The relationship between the angle of elevation and the angle of depression lies in their complementary nature in right-angle trigonometry. Both angles are measured with respect to a horizontal line and can be used to calculate distances or heights in various real-world applications.
- Angle of Elevation: This is the angle formed between the horizontal line and the line of sight when looking upwards at an object. For instance, if you are standing on the ground and looking up at a tree, the angle between the horizontal ground and your line of sight to the top of the tree is the angle of elevation.
- Angle of Depression: This is the angle formed between the horizontal line and the line of sight when looking downwards at an object. For example, if you are standing at the top of a building and looking down at a car parked on the street, the angle between the horizontal line at your eye level and your line of sight to the car is the angle of depression.
Relationship:
- The key relationship between the angle of elevation and the angle of depression is that they are equal when measured from two points of observation at different heights. This is because they form alternate interior angles with a horizontal line when considering the observer’s line of sight. In other words, if you draw a horizontal line from the observer’s point and extend a line of sight upward or downward, the angle of elevation from one point equals the angle of depression from the other point.
Example of Application:
Imagine you are standing 50 meters away from the base of a building, and you want to calculate the height of the building. You measure the angle of elevation to the top of the building as 30°. Using trigonometry, you can calculate the height of the building using the tangent function:
[
\text{tan}(\text{angle of elevation}) = \frac{\text{opposite (height)}}{\text{adjacent (distance)}}
]
Thus, the height ( h ) of the building can be found as:
[
h = 50 \times \tan(30^\circ) \approx 50 \times 0.577 = 28.85 \text{ meters}
]
This shows how the angle of elevation is used to solve real-world problems involving heights and distances.