JEE Maths Notes: Chapter on Trigonometric Properties

1. Introduction to Trigonometric Properties

Trigonometric properties are fundamental principles used to relate the various trigonometric functions of angles. These properties help in simplifying expressions, solving trigonometric equations, and proving identities. In this chapter, we will explore the various essential properties of trigonometric functions, their interrelations, and how they apply in solving mathematical problems.

2. Trigonometric Identity Properties

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable in a specified domain. Some important properties include:

• Basic Pythagorean Identity: sin² θ + cos² θ = 1
• Secant and Tangent Identity: sec² θ = 1 + tan² θ
• Cosecant and Cotangent Identity: csc² θ = 1 + cot² θ
• Reciprocal Identities: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ

These identities serve as the foundation for simplifying trigonometric expressions and solving complex equations.

3. Addition and Subtraction Formulas

The addition and subtraction formulas are crucial in simplifying trigonometric expressions that involve the sum or difference of angles. The basic addition formulas are:

• sin (A + B) = sin A cos B + cos A sin B
• cos (A + B) = cos A cos B - sin A sin B
• tan (A + B) = (tan A + tan B) / (1 - tan A tan B)

These formulas also have corresponding subtraction versions, such as:

• sin (A - B) = sin A cos B - cos A sin B
• cos (A - B) = cos A cos B + sin A sin B
• tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

These formulas are particularly useful in simplifying expressions and solving trigonometric equations involving sums and differences of angles.

4. Double Angle and Half Angle Formulas

Double angle formulas express trigonometric functions of double angles (2θ) in terms of single angles (θ). The half angle formulas help in expressing functions of half angles (θ/2). These formulas are as follows:

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos² θ - sin² θ
• tan 2θ = (2 tan θ) / (1 - tan² θ)

The half angle formulas are:

• sin (θ/2) = ±√((1 - cos θ) / 2)
• cos (θ/2) = ±√((1 + cos θ) / 2)
• tan (θ/2) = ±√((1 - cos θ) / (1 + cos θ))

These formulas help in solving problems involving trigonometric functions of multiples or fractions of angles, making them essential for problem-solving in trigonometry.

5. Product-to-Sum and Sum-to-Product Formulas

These formulas allow you to convert products of trigonometric functions into sums or differences and vice versa. They are useful in simplifying expressions and solving trigonometric integrals and equations.

• Product-to-Sum Formulas:
  • sin A sin B = (1/2)[cos(A - B) - cos(A + B)]
  • cos A cos B = (1/2)[cos(A - B) + cos(A + B)]
  • sin A cos B = (1/2)[sin(A + B) + sin(A - B)]
• Sum-to-Product Formulas:
  • sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2)
  • cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2)
  • sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2)

These formulas are useful for simplifying complex trigonometric expressions, particularly when working with products and sums of sines and cosines.

6. Trigonometric Functions of Multiple Angles

The trigonometric functions of multiple angles are expressions involving integer multiples of a given angle. These functions are crucial when working with periodic phenomena and oscillations. Examples of multiple angle formulas include:

• sin nθ, cos nθ, and tan nθ can be