Trigonometric Identities Lecture Notes

Overview

Trigonometric identities are crucial in simplifying expressions and solving equations involving trigonometric functions. They are vital in fields such as geometry, engineering, and physics.

Key Topics

1. Definition of Trigonometric Identities

   • Equalities involving trigonometric functions, valid for all variable values.
   • Simplify complex expressions and solve equations easily.
   • Six basic trigonometric ratios: sine, cosine, tangent, cosecant, secant, cotangent.

2. List of Trigonometric Identities

   • Reciprocal Identities:
     • sin = 1/cosec, cosec = 1/sin
     • cos = 1/sec, sec = 1/cos
     • tan = 1/cot, cot = 1/tan
   • Pythagorean Identities:
     • sin² + cos² = 1
     • 1 + tan² = sec²
     • cosec² = 1 + cot²
   • Ratio Identities:
     • tan = sin/cos
     • cot = cos/sin
   • Opposite Angles Identities:
     • sin(-θ) = -sinθ
     • cos(-θ) = cosθ
     • tan(-θ) = -tanθ
   • Complementary Angles Identities:
     • sin(90° - θ) = cosθ
     • cos(90° - θ) = sinθ
   • Supplementary Angles Identities:
     • sin(180° - θ) = sinθ
     • cos(180° - θ) = -cosθ
   • Periodicity:
     • sin(n×360° + θ) = sinθ
     • cos(n×360° + θ) = cosθ
     • tan(n×180° + θ) = tanθ
   • Sum and Difference Identities:
     • sin(A+B) = sinA cosB + cosA sinB
     • cos(A+B) = cosA cosB - sinA sinB
   • Double Angle Identities:
     • sin2θ = 2sinθ cosθ
     • cos2θ = cos²θ - sin²θ
   • Half-Angle Formulas:
   • Product-Sum Identities:
   • Triple Angle Formulas:

3. Proof of Trigonometric Identities

   • Use right-angled triangles to prove basic identities.

4. Relation Between Angles and Sides of Triangle

   • Sine Rule states the ratio of a side to the sine of its opposite angle is constant for all sides and angles of the triangle.
   • Cosine Rule provides a relationship between the lengths of a triangle's sides and its angles.
   • Tangent Rule uses tan to relate sides and angles.

5. Examples and Practice Problems

   • Several proofs and example problems illustrate the application of these identities.
   • Practice problems to test understanding and application.

Conclusion

Trigonometric identities are powerful tools in mathematical calculations, providing a foundational understanding necessary for solving complex problems in mathematics and science.