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Class 10 · Chapter 9 · Mathematics

Some Applications of Trigonometry Cheat Sheet

Formulae

Height from angle of elevation

tan⁡θ=heighthorizontal distance\tan\theta = \frac{\text{height}}{\text{horizontal distance}}

Used when the object is above the observer’s eye level.

Angle of elevation and depression

tan⁡θ=hd\tan\theta = \frac{h}{d}

h = vertical height, d = horizontal distance, θ = angle of elevation or depression; the two angles are equal for the same line of sight.

Using the standard angles

tan⁡30∘=13,tan⁡45∘=1,tan⁡60∘=3\tan 30^\circ = \frac{1}{\sqrt{3}}, \quad \tan 45^\circ = 1, \quad \tan 60^\circ = \sqrt{3}

Most problems reduce to a right triangle with one of these angles.

Definitions

Line of sight
The straight line from the eye of the observer to the object being viewed.
Angle of elevation
The angle between the line of sight and the horizontal when the object is above the horizontal level of the eye.
Angle of depression
The angle between the line of sight and the horizontal when the object is below the horizontal level of the eye.
Horizontal line
A line parallel to the ground through the observer’s eye.

Key Points

  • Trigonometry is used to find heights and distances that cannot be measured directly.
  • The angle of elevation is measured upwards from the horizontal; the angle of depression is measured downwards.
  • The angle of elevation from A to B equals the angle of depression from B to A (alternate angles).
  • In these problems the unknown height or distance is found from tan θ = perpendicular/base.
  • Common angles used are 30°, 45° and 60°, whose tangent values are 1/√3, 1 and √3.
  • A single right triangle gives one equation; two right triangles may be needed when the object has two observation points.
  • Diagrams are drawn with the horizontal ground, the vertical object and the line of sight clearly marked.