Introduction to Height and Distance (original) (raw)
Last Updated : 1 Jun, 2026
Height is the measurement of an item in the vertical direction, whereas distance is the measurement of an object in the horizontal direction. Heights and Distances are the real-life applications of trigonometry which is useful to astronomers, navigators, architects, surveyors, etc., in solving problems related to heights and distances.
In height and distance, we use trigonometric concepts to find the height and distance of various objects.
Height and Distance in Trigonometry
Various terminologies that help understand Height and Distance are,
- **Line of Sight: It is the line drawn from the eye of an observer to the point in the object viewed by the observer.
- **Angle of Elevation: The angle between the horizontal and the line of sight joining an observation point to an elevated object is called the angle of elevation.
- **Angle of Depression: The angle between the horizontal and the line of sight joining an observation point to an object below the horizontal level is called the angle of depression.
How to Find Heights and Distances?
Trigonometric ratios are used to measure the heights and distances of different objects. For example, in the above figure, a person is looking at the top of the tree; the angle from the eye level to the top of the tree is called the angle of elevation, and similarly, the angle from the top of the tree to the eyes of the person is called the angle of depression.
If the height of the person and their distance from the tree are known, we can easily calculate the height of the tree using various trigonometric ratios.
Trigonometric Ratios Table
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). These ratios represent the relationships between the sides of a right-angled triangle with respect to a given angle θ. The trigonometric ratios for a given angle θ are shown below:
| **Trigonometric Ratio | **Definition |
|---|---|
| sin θ | Perpendicular / Hypotenuse |
| cos θ | Base / Hypotenuse |
| tan θ | Perpendicular / Base |
| sec θ | Hypotenuse / Base |
| cosec θ | Hypotenuse / Perpendicular |
| cot θ | Base / Perpendicular |
The value of trigonometric ratios for different angles is very useful for solving Height and Distance problems. Thus, it is advised to learn the values of trigonometric ratios for different angles. The value of various trigonometric ratios can be learned using the trigonometric table provided below.
| Angles (In Degrees) | 0 | 30 | 45 | 60 | 90 | 180 | 270 |
|---|---|---|---|---|---|---|---|
| Angles (In Radians) | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Not Defined | 0 | Not Defined |
| cot | Not Defined | √3 | 1 | 1/√3 | 0 | Not Defined | 0 |
| cosec | Not Defined | 2 | √2 | 2/√3 | 1 | Not Defined | -1 |
| sec | 1 | 2/√3 | √2 | 2 | Not Defined | -1 | Not Defined |