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Schooling · Class 10 · Mathematics · Some Applications of Trigonometry · Chapter on a page

CBSE Class 10 · Mathematics · NCERT Mathematics

Some Applications of Trigonometry: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Some Applications of Trigonometry: Key ideas, Formulas and facts, Watch out forKey ideasThe angle of elevation is the angle…The angle of depression is the angle…When an observer looks down at an…In any right triangle, tan θ =…Drawing a clear, labelled diagram is…Most height-and-distance problems…Problems may involve two positions…Formulas and factstan θ = perpendicular / base — use…sin θ = perpendicular / hypotenuse…cos θ = base / hypotenuse — use when…tan 30° = 1/√3sin 30° = 1/2cos 30° = √3/2Angle of depression from A to B =…Watch out forConfusing angle of elevation with…Using the wrong trigonometric ratioForgetting to add the observer's…Placing the right angle incorrectly…Rounding too early in calculationsSome Applications ofTrigonometry
The mind map as a list
  • Key ideas
    • The angle of elevation is the angle…
    • The angle of depression is the angle…
    • When an observer looks down at an…
    • In any right triangle, tan θ =…
    • Drawing a clear, labelled diagram is…
    • Most height-and-distance problems…
    • Problems may involve two positions…
  • Formulas and facts
    • tan θ = perpendicular / base — use…
    • sin θ = perpendicular / hypotenuse…
    • cos θ = base / hypotenuse — use when…
    • tan 30° = 1/√3
    • sin 30° = 1/2
    • cos 30° = √3/2
    • Angle of depression from A to B =…
  • Watch out for
    • Confusing angle of elevation with…
    • Using the wrong trigonometric ratio
    • Forgetting to add the observer's…
    • Placing the right angle incorrectly…
    • Rounding too early in calculations

What this chapter is about

This chapter shows how the trigonometric ratios you learnt earlier can solve real-world problems involving heights and distances. When direct measurement is impossible — finding the height of a tall building, the width of a river, or the distance of a ship from shore — trigonometry provides a practical method using angles and one known length.

Key ideas

  • The angle of elevation is the angle between the horizontal line of sight and the line of sight directed upward to an object above the observer.
  • The angle of depression is the angle between the horizontal line of sight and the line of sight directed downward to an object below the observer.
  • When an observer looks down at an object, the angle of depression from the observer equals the angle of elevation from the object to the observer (alternate angles with a horizontal line).
  • In any right triangle, tan θ = opposite side / adjacent side, sin θ = opposite side / hypotenuse, and cos θ = adjacent side / hypotenuse.
  • Drawing a clear, labelled diagram is essential: mark the right angle, the known angle, the known length, and the unknown length.
  • Most height-and-distance problems reduce to finding one side of a right triangle when another side and an acute angle are known.
  • Problems may involve two positions of observation (moving closer or farther) to set up two equations and eliminate an unknown.

Formulas and facts to remember

  • tan θ = perpendicular / base — use when height and horizontal distance are involved.
  • sin θ = perpendicular / hypotenuse — use when height and slant distance are involved.
  • cos θ = base / hypotenuse — use when horizontal and slant distances are involved.
  • Standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
  • Standard values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2.
  • Standard values: cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2.
  • Angle of depression from A to B = Angle of elevation from B to A (when horizontal lines through A and B are parallel).

Common mistakes

  • Confusing angle of elevation with angle of depression → Remember elevation is looking up, depression is looking down; always draw the horizontal reference line first.
  • Using the wrong trigonometric ratio → Identify which sides you have (opposite, adjacent, hypotenuse relative to the given angle) before choosing sin, cos or tan.
  • Forgetting to add the observer's height when standing above ground level → If a person of height 1.5 m observes a tower, the calculated height is from eye level; add 1.5 m for total tower height.
  • Placing the right angle incorrectly in the diagram → The right angle is at the base of the vertical object (tower, building), not at the observer's position.
  • Rounding too early in calculations → Keep √3 or √2 in your working until the final step to avoid accumulating errors.

Quick revision

  • Angle of elevation: observer looks upward; angle of depression: observer looks downward.
  • For most problems, tan θ is your primary tool since you deal with height and horizontal distance.
  • Always draw the diagram first — label all known and unknown quantities.
  • Use tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 for quick calculations.
  • When two observation points are given, form two equations and solve simultaneously.
  • The observer's own height must be added to find the full height of tall objects.