Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. These functions play a fundamental role in geometry, physics, engineering, and various mathematical applications. Here are key concepts related to trigonometric functions:

1. Basic Definitions:

  • Sine Function (sin):

    • Defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
    • sin(θ)=OppositeHypotenuse
  • Cosine Function (cos):

    • Defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
    • cos(θ)=AdjacentHypotenuse
  • Tangent Function (tan):

    • Defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.
    • tan(θ)=OppositeAdjacent
  • Cosecant Function (csc), Secant Function (sec), Cotangent Function (cot):

    • Reciprocal functions of sine, cosine, and tangent, respectively.
    • csc(θ)=1sin(θ)
    • sec(θ)=1cos(θ)
    • cot(θ)=1tan(θ)

2. Unit Circle:

  • Trigonometric functions can be defined on the unit circle, where the radius is 1.
  • The coordinates of a point on the unit circle correspond to the values of sin and cos for a specific angle.

3. Periodicity:

  • All trigonometric functions are periodic with a period of 360 or 2π radians.
  • sin(θ+360)=sin(θ) and cos(θ+2π)=cos(θ)

4. Trigonometric Identities:

  • Pythagorean Identity:
    • sin2(θ)+cos2(θ)=1
  • Reciprocal Identities:
    • csc2(θ)=sin2(θ)+1
    • sec2(θ)=cos2(θ)+1
    • cot2(θ)=1+tan2(θ)
  • Quotient Identities:
    • tan(θ)=sin(θ)cos(θ)

5. Trigonometric Formulas:

  • Sum and Difference Formulas:
    • sin(α±β)=sin(α)cos(β)±cos(α)sin(β)
    • cos(α±β)=cos(α)cos(β)sin(α)sin(β)
  • Double-Angle Formulas:
    • sin(2θ)=2sin(θ)cos(θ)
    • cos(2θ)=cos2(θ)sin2(θ)
  • Half-Angle Formulas:
    • sin(θ2)=±1cos(θ)2
  • cos(θ2)=±1+cos(θ)2