Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant. These identities are true for all values of the variables where both sides of the equation are defined.

Basic Trigonometric Identities

  1. Reciprocal Identities:

    • csc(θ) = 1/sin(θ)
    • sec(θ) = 1/cos(θ)
    • cot(θ) = 1/tan(θ)

  2. Quotient Identities:

    • tan(θ) = sin(θ)/cos(θ)
    • cot(θ) = cos(θ)/sin(θ)

  3. Pythagorean Identities:

    • [sin(θ)]² + [cos(θ)]² = 1
    • 1 + [tan(θ)]² = [sec(θ)]²
    • 1 + [cot(θ)]² = [csc(θ)]²

Co-Function Identities

These identities relate functions of complementary angles (two angles whose measures add up to 90 degrees or π/2 radians):

  • sin(π/2 - θ) = cos(θ)
  • cos(π/2 - θ) = sin(θ)
  • tan(π/2 - θ) = cot(θ)
  • cot(π/2 - θ) = tan(θ)
  • sec(π/2 - θ) = csc(θ)
  • csc(π/2 - θ) = sec(θ)

Double-Angle and Half-Angle Identities

These identities express trigonometric functions of double angles or half angles in terms of functions of the original angles:

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

Applications

Trigonometric identities are used in various areas of mathematics, including algebra, calculus, and complex analysis. They also have applications in physics and engineering, particularly in wave analysis and signal processing.