# Height and Distance

1. Introduction:

• The concept of height and distance in trigonometry involves using trigonometric functions to solve problems related to measuring heights, distances, and angles. This is particularly useful in real-world scenarios where direct measurements may not be feasible.

2. Basic Trigonometric Ratios:

• Sine, Cosine, and Tangent:
• In a right-angled triangle, the sine ($\mathrm{sin}$), cosine ($\mathrm{cos}$), and tangent ($\mathrm{tan}$) of an angle are defined as follows:
• $\mathrm{sin}\left(\theta \right)=\frac{\text{Opposite}}{\text{Hypotenuse}}$
• $\mathrm{cos}\left(\theta \right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• $\mathrm{tan}\left(\theta \right)=\frac{\text{Opposite}}{\text{Adjacent}}$

3. Angle of Elevation and Depression:

• The angle of elevation is the angle formed between the line of sight and the horizontal when looking upward. The angle of depression is the angle formed when looking downward. These concepts are essential in height and distance problems.

4. Solving Problems Involving Height and Distance:

• Problem-Solving Steps:
• Identify the right-angled triangle formed by the observer, the object, and the horizontal line.
• Label the sides of the triangle as opposite, adjacent, and hypotenuse.
• Choose the appropriate trigonometric ratio based on the information given.
• Set up and solve the trigonometric equation to find the unknown quantity (height or distance).

5. Example Problem:

• Problem:
• A person standing at a distance of 20 meters from the base of a tower observes the angle of elevation to the top of the tower as $3{0}^{\circ }$. Find the height of the tower.
• Solution:
• Identify the right-angled triangle formed by the person, the top of the tower, and the base. Use the tangent ratio: $\mathrm{tan}\left(3{0}^{\circ }\right)=\frac{\text{height}}{20}$ to find the height.

6. Pythagorean Theorem in Height and Distance:

• In some problems, the Pythagorean Theorem (${a}^{2}+{b}^{2}={c}^{2}$) may be applicable, where $a$ and $b$ are the legs of the right-angled triangle, and $c$ is the hypotenuse.

7. Multiple Heights and Distances:

• Problems may involve multiple objects or points of observation. In such cases, creating and solving separate triangles for each observation point is common.

8. Applications in Surveying:

• Trigonometry is extensively used in surveying to measure distances and elevations. Theodolites and total stations employ trigonometric principles for accurate measurements.

9. Real-world Applications:

• Building Construction:
• Trigonometry is used to calculate heights and distances in construction projects.
• Astronomy:
• In astronomy, trigonometry helps measure distances to celestial objects based on angles of observation.
• Geography:
• Height and distance calculations are crucial in geography, especially when mapping and measuring terrain.

10. Summary:

• Height and distance problems in trigonometry involve using trigonometric ratios to find unknown quantities, such as heights or distances, based on angles of observation.