Some Applications of Trigonometry

When we look at an object, the straight line drawn from our eye to the object is called the line of sight. The horizontal line through the eye is called the horizontal line. The angle between the line of sight and the horizontal line is what we measure in this chapter.

• If the object is above the horizontal level, we raise our head to see it. The angle formed is the angle of elevation.
• If the object is below the horizontal level, we lower our head to see it. The angle formed is the angle of depression.

A very useful fact: the angle of depression of an object from an observer is equal to the angle of elevation of the observer from that object. This is because the two horizontal lines are parallel, and these angles are alternate interior angles. This lets us mark the elevation angle at the base of the object even when the problem gives a depression from the top.

Worked example. Suppose the angle of elevation of the top of a vertical pole from a point on the ground 15 m from its foot is 45°. If the height is h, then in the right triangle, tan 45° = h / 15. Since tan 45° = 1, we get h = 15 m. So the pole is 15 m tall.

In an elevation problem the vertical object (tower, pole, tree) is the side opposite the angle, and the horizontal ground distance is the side adjacent to the angle. When both of these appear, the correct ratio is the tangent:

tan θ = height (opposite) / horizontal distance (adjacent).

Choose sin or cos instead when the problem involves the slant line itself (the hypotenuse), such as a rope, string, or ladder length.

Worked example. The angle of elevation of the top of a tower from a point on the ground, 30 m away from the foot of the tower, is 30°. Find the height. Let the height be h. In the right triangle, tan 30° = h / 30. Since tan 30° = 1/√3, we have h = 30 × (1/√3) = 30/√3 = 10√3 m ≈ 17.32 m.

Always draw the figure, label the right angle at the foot of the vertical object, mark the given angle at the observer's point, and only then pick the ratio that connects the known and unknown sides.

When an observer at the top of a tower or building looks down at an object on the ground, the angle of depression is measured from the horizontal line at the observer's eye down to the line of sight. To solve, use the parallel-line fact: the angle of depression equals the angle of elevation of the observer from the object. So we can mark the same angle at the ground-level object and work with a familiar right triangle.

Worked example. From the top of a building 40 m high, the angle of depression of a car standing on the ground is 45°. Find the distance of the car from the foot of the building. The depression angle 45° equals the angle of elevation of the top from the car. If d is the required distance, then tan 45° = 40 / d. Since tan 45° = 1, we get d = 40 m.

A second example. From the same 40 m building the depression of another car is 30°. Then tan 30° = 40 / d, so d = 40 / tan 30° = 40 × √3 = 40√3 m ≈ 69.28 m. The smaller the angle of depression, the farther the object.

Every problem in this chapter is built from the three standard angles 30°, 45° and 60°. Memorising their sine, cosine and tangent values makes the calculation exact and quick.

• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

To decide which ratio to use, identify which sides you know and want:

1. Opposite and adjacent (the two legs) → use tan θ.
2. Opposite and hypotenuse → use sin θ.
3. Adjacent and hypotenuse → use cos θ.

Worked example. A ladder 10 m long rests against a wall and makes an angle of 60° with the ground. Find the height on the wall it reaches. Here the ladder is the hypotenuse and the height is opposite the 60° angle, so use sine: height = 10 × sin 60° = 10 × (√3/2) = 5√3 m ≈ 8.66 m. To find how far the foot is from the wall we would use cosine: 10 × cos 60° = 10 × (1/2) = 5 m.

Many board questions involve two angles: for example the Sun's shadow at two different elevations, two ships seen from a lighthouse, or a tower fixed on top of a building. The method is to build two right-triangle equations that share a common side (usually the same height or the same horizontal distance) and then combine them.

Worked example. The shadow of a tower is 20 m longer when the Sun's elevation is 30° than when it is 60°. Find the height of the tower. Let the height be h. When the elevation is 60°, the shadow is h / tan 60° = h/√3. When it is 30°, the shadow is h / tan 30° = h√3. The difference is:

h√3 − h/√3 = 20

Taking a common denominator, (3h − h)/√3 = 20, so 2h/√3 = 20, giving h = 10√3 m ≈ 17.32 m.

The same idea solves a tower-on-a-building problem: the lower angle gives the horizontal distance, and the higher angle gives the total height; subtracting the building height gives the tower's height. Careful labelling of the shared side is the key skill.