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Mathematics: Class 10 — Chapter 8: Introduction to Trigonometry

8.1 Introduction

Concept and Origin

  • Trigonometry is the study of relationships between the sides and angles of a triangle.
  • Derived from Greek words: tri (three), gon (sides), and metron (measure).
  • Historically used by astronomers to find distances to stars and planets, and widely used today in engineering and physical sciences.
8.2 Trigonometric Ratios

Definitions of Ratios

For a right-angled triangle ABC (right-angled at B) with respect to acute angle A:

  • sin A = Side opposite to angle A / Hypotenuse = BC / AC
  • cos A = Side adjacent to angle A / Hypotenuse = AB / AC
  • tan A = Side opposite to angle A / Side adjacent to angle A = BC / AB = sin A / cos A
  • cosec A = 1 / sin A = Hypotenuse / Side opposite to angle A = AC / BC
  • sec A = 1 / cos A = Hypotenuse / Side adjacent to angle A = AC / AB
  • cot A = 1 / tan A = Side adjacent to angle A / Side opposite to angle A = AB / BC = cos A / sin A

Important Properties

  • The values of trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.
  • Since the hypotenuse is the longest side in a right triangle, the value of sin A or cos A is always less than or equal to 1.
8.3 Trigonometric Ratios of Specific Angles

Standard Values Table

Ratio (θ)0°30°45°60°90°
sin θ01/21/√2√3/21
cos θ1√3/21/√21/20
tan θ01/√31√3Not defined
cosec θNot defined2√22/√31
sec θ12/√3√22Not defined
cot θNot defined√311/√30
8.4 Trigonometric Identities

Fundamental Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle(s) involved.

  • cos² A + sin² A = 1 (for 0° ≤ A ≤ 90°)
  • 1 + tan² A = sec² A (for 0° ≤ A < 90°)
  • cot² A + 1 = cosec² A (for 0° < A ≤ 90°)