CBSE Class 10 Mathematics Some Applications of Trigonometry Notes Set 04

Line of Sight, Elevation and Depression

The line of sight is the straight line from the observer's eye to the object being looked at. The horizontal line through the eye is the reference.
Angle of elevation: the angle between the horizontal and the line of sight when the object is above the observer's eye level. Angle of depression: the angle between the horizontal and the line of sight when the object is below eye level.
The angle of elevation of the observer as seen from the object equals the angle of depression of the object as seen from the observer, because they are alternate angles between parallel horizontal lines.

Method for Height and Distance Problems

Step 1: read the statement and draw a neat figure, marking the vertical height, the horizontal distance and the given angles.
Step 2: find a right triangle that contains the unknown and one known quantity. Choose the ratio that links the two sides: \( \tan\theta = \frac{\text{height}}{\text{horizontal distance}} \) is the most common, with sin and cos used when the slant length (ladder, rope, slope, line of sight) is involved.
Step 3: when the observer moves or two objects are seen, write one equation for each position and eliminate the common unknown. Standard values to remember: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
Step 4: take √3 = 1.732 unless told otherwise and give the answer with units.

Standard Results Worth Remembering

Observer at the top of a tower of height h sees two points on the ground on the same side with depressions α and β (β larger, nearer point). The gap between them is \( h(\cot\alpha - \cot\beta) \).
Tower of height H, observer moves a distance d towards it and the elevation changes from α to β: \( H = \frac{d}{\cot\alpha - \cot\beta} \). For 30° to 60° this gives \( H = \frac{d\sqrt3}{2} \).
If the elevations of the top of a tower from two points at distances a and b from its foot on the same line (on the same side) are complementary, the height is \( \sqrt{ab} \).
Two points on opposite sides of a point directly below an aeroplane, 1 unit apart, with depressions α and β: height \( \frac{\tan\alpha\tan\beta}{\tan\alpha+\tan\beta} \).

Worked Examples

Example 1: the shadow of a pole is √3 times its length. Then tan θ = 1/√3, so the sun's elevation is 30°.
Example 2: from the top of a 120 m tower the angle of depression of a car is 30°. Horizontal distance = 120 cot 30° = 120√3 ≈ 207.84 m.
Example 3: a hill and a tower stand on level ground; the tower (50 m) subtends 30° at the hill's foot and the hill subtends 60° at the tower's foot. Distance between the feet = 50 cot 30° = 50√3 m, so hill height = 50√3 tan 60° = 150 m.
Example 4: from a lighthouse 100 m high the depression of a ship changes from 30° to 45° as it sails towards the base. Distance covered = 100(√3 − 1) ≈ 73.2 m.