Some Applications of Trigonometry — Study Notes

1. Introduction

This chapter applies the trigonometric ratios learned in the previous chapter to real-life situations involving heights and distances. It introduces two crucial concepts: the line of sight and the angle of elevation and angle of depression. Using a right triangle drawn through the observer's eye and the object being observed, trigonometric ratios allow us to compute heights of towers, buildings, and mountains, as well as horizontal distances, without physically measuring them.

The key idea is that whenever an object is viewed from a certain point, the line from the observer's eye to the object makes an angle with the horizontal line. If the object is above the horizontal, the angle is the angle of elevation; if it is below, the angle of depression. Along with the trigonometric ratios of standard angles, these angles convert the problem into a simple right-triangle computation.

This chapter is a favourite in board examinations because it combines real-world meaning with straightforward computation. Questions typically present a diagram in words, ask for a height or a distance, and require students to draw, label, and solve the appropriate right triangle using tan, sin, or cos.

2. Line of Sight, Angle of Elevation, and Angle of Depression

Line of Sight

The line of sight is the straight line from the observer's eye to the object being viewed.

Angle of Elevation

If the object is above the level of the observer's eye, the angle between the horizontal line and the line of sight, measured upwards, is called the angle of elevation.

Angle of Depression

If the object is below the level of the observer's eye, the angle between the horizontal line and the line of sight, measured downwards, is called the angle of depression.

A key property: the angle of depression from the observer to an object equals the angle of elevation from the object to the observer, because the horizontal line and the line of sight form alternate interior angles with a parallel line structure.

3. The Basic Setup

Consider a tower of height h standing on level ground. From a point at a horizontal distance d from the foot of the tower, if the angle of elevation of the top of the tower is theta, then: tan theta = h/d

From this, we get: h = d tan theta d = h/tan theta = h cot theta

Example: If the angle of elevation of the top of a tower is 30 degrees from a point 60 m away, then: h = 60 x tan 30 = 60 x 1/sqrt(3) = 60/sqrt(3) = 20 sqrt(3) m.

This simple relation tan theta = height/distance is the heart of almost every problem in this chapter.

4. Two-Observer and Multi-Observation Problems

Some questions involve observing the same object from two different positions, leading to a system of equations.

Example: Tower Observed from Two Points

A tower is observed from two points on the same side of the tower at distances d1 and d2, with angles of elevation alpha and beta. Then: h = d1 tan alpha = d2 tan beta

The difference d1 - d2 (when the angles are measured from two positions along the line) gives an equation that can be solved for h and the distances.

Example: Object and Its Shadow

If the sun's angle of elevation is theta, and a vertical object of height h casts a shadow of length s, then: tan theta = h/s

Problems involving the elevation of the sun are solved with this single relation. When the sun's elevation changes, the shadow length changes accordingly, and comparing the two situations gives a solvable system.

5. Angle of Depression Problems

For an observer at a height h looking down at an object at horizontal distance d, with angle of depression theta: tan theta = h/d

Example: From the top of a 30 m building, the angle of depression of a car on the road is 45 degrees. Then: 30/d = tan 45 = 1, so d = 30 m.

The angle of depression is always measured from the horizontal downwards, and it equals the angle of elevation of the observer as seen from the object.

6. Solved Strategy for Word Problems

1. Read the problem and identify the observer, the object, and the horizontal level.
2. Draw a neat right triangle with the vertical height, horizontal distance, and line of sight clearly marked.
3. Mark the given angle and the given length; identify the unknown.
4. Choose the appropriate ratio (usually tan = height/distance).
5. Substitute the standard values of trigonometric functions and solve.
6. Write the answer with proper units, rationalising surds where needed.

Most problems reduce to one or two applications of tan theta = h/d, and a correct labelled figure is worth most of the marks.

Quick Revision Tables

Table 1: Core Relations

Table 2: Common Standard Values Used

Mind Map

Important Diagrams (SVG)

Diagram 1: Angle of Elevation and Depression

Diagram 2: Shadow and Elevation of the Sun

Common Mistakes

1. Measuring the angle of elevation from the vertical instead of the horizontal; the angle is always with the horizontal line.
2. Using the distance between the observer's eye and the object (the line of sight) instead of the horizontal distance, mixing up the hypotenuse and the base.
3. Forgetting to add the height of the observer to the computed height when the object's total height is asked and the observer is standing at ground level with a known eye height.
4. Using sin or cos where tan is appropriate, or vice versa, because the figure is not labelled clearly.
5. Ignoring the given units and reporting answers without units or mixing metres and centimetres.
6. In two-observer problems, subtracting distances the wrong way, or equating heights without using the correct tangent relations.
7. Leaving answers as unsimplified surds, e.g., writing 60/sqrt(3) instead of rationalising to 20 sqrt(3).

Exam Tips

1. Draw a labelled diagram first; nearly all marks in these problems depend on a correct figure.
2. Decide which quantity is the hypotenuse and which is the base, and choose tan theta = h/d when the height and distance are involved.
3. When the observer's height matters, add it explicitly at the end of the calculation.
4. Memorise the standard tangent values, since tan is used in most height-and-distance problems.
5. For two-position problems, write both equations h = d1 tan alpha and h = d2 tan beta and equate them.
6. Rationalise surd answers, e.g., write 20 sqrt(3) instead of 60/sqrt(3).
7. Practise the "sun elevation and shadow length" variant, which is a frequent short-answer question.

Conclusion

The applications of trigonometry transform abstract ratios into a practical tool for measuring the world. With just the angle of elevation or depression and the tangent ratio, students can determine the heights of buildings and towers, the distance to an object, and the elevation of the sun. The systematic method of drawing, labelling, and solving a right triangle makes these problems reliable and scoring in the examination. This chapter also reinforces the real-world value of mathematics, showing how a few trigonometric ratios enable measurements that would otherwise require sophisticated instruments, and it prepares students for the deeper applications of trigonometry in science and engineering.