INTRODUCTION TO TRIGONOMETRY

1. Introduction and Origins

• Definition: Trigonometry is the branch of mathematics dealing with the relationships between the sides and angles of a triangle.
• Etymology: The word is derived from the Greek words 'tri' (three), 'gon' (sides), and 'metron' (measure).
• Historical Context:
  • Early records exist from Egypt and Babylon.
  • Ancient astronomers used it to calculate distances to stars and planets.
  • The concept of "sine" was first used in the work Aryabhatiyam by Aryabhata (A.D. 500). The term evolved from ardha-jya to jiva, then to the Latin sinus.
• Modern Application: It forms the basis for technologically advanced methods in Engineering and Physical Sciences.

2. Trigonometric Ratios

Trigonometric ratios express the relationship between an acute angle in a right triangle and the lengths of its sides.

Definitions based on a Right Triangle ABC (angle B = 90°):

For an acute angle A:

• Sine (sin A): Ratio of the side opposite to angle A to the hypotenuse.
  sin A = Opposite / Hypotenuse
• Cosine (cos A): Ratio of the side adjacent to angle A to the hypotenuse.
  cos A = Adjacent / Hypotenuse
• Tangent (tan A): Ratio of the side opposite to angle A to the side adjacent to angle A.
  tan A = Opposite / Adjacent

Reciprocal Ratios:

• Cosecant (cosec A): Reciprocal of sine.
  cosec A = 1 / sin A = Hypotenuse / Opposite
• Secant (sec A): Reciprocal of cosine.
  sec A = 1 / cos A = Hypotenuse / Adjacent
• Cotangent (cot A): Reciprocal of tangent.
  cot A = 1 / tan A = Adjacent / Opposite

Key Properties:

• Quotient Relations: tan A = sin A / cos A and cot A = cos A / sin A.
• Invariance: The values of trigonometric ratios depend only on the angle, not on the size of the triangle (proven via similarity of triangles).
• Range: Since the hypotenuse is the longest side, the values of sin A and cos A are always less than or equal to 1. Consequently, sec A and cosec A are always greater than or equal to 1.
• Notation: (sin A)² is written as sin² A. Note that sin^-1 A is different from (sin A)^-1.

3. Trigonometric Ratios of Specific Angles

Values for specific angles (0°, 30°, 45°, 60°, 90°) are derived using geometry (equilateral and isosceles right triangles).

Observations:

• As angle A increases from 0° to 90°, sin A increases from 0 to 1.
• As angle A increases from 0° to 90°, cos A decreases from 1 to 0.
• tan 90°, sec 90°, cosec 0°, and cot 0° are not defined.

4. Trigonometric Identities

A trigonometric identity is an equation involving trigonometric ratios of an angle that is true for all values of the angle.

Utility of Identities:

• These identities allow the expression of any trigonometric ratio in terms of another.
• Example: If sin A is known, cos A can be found using √(1 - sin² A).
• They are used to simplify trigonometric expressions and prove other equations.

Summary derived from Chapter 8 "Introduction to Trigonometry".