Trigonometric Ratios

I. Introduction: Trigonometric ratios are fundamental relationships between the angles and sides of a right-angled triangle. They provide a framework for understanding the geometric properties of triangles and have widespread applications in mathematics and various scientific disciplines.

II. Trigonometric Ratios in a Right-Angled Triangle:

  1. Sine (sinθ):

    • Definition: sinθ=Opposite sideHypotenuse
    • Relationship: sinθ=ac
  2. Cosine (cosθ):

    • Definition: cosθ=Adjacent sideHypotenuse
    • Relationship: cosθ=bc
  3. Tangent (tanθ):

    • Definition: tanθ=Opposite sideAdjacent side
    • Relationship: tanθ=ab

III. Reciprocal Trigonometric Ratios:

  1. Cosecant (cscθ):

    • Definition: cscθ=1sinθ=HypotenuseOpposite side
  2. Secant (secθ):

    • Definition: secθ=1cosθ=HypotenuseAdjacent side
  3. Cotangent (cotθ):

    • Definition: cotθ=1tanθ=Adjacent sideOpposite side

IV. Unit Circle Representation:

  1. Coordinates on the Unit Circle:
    • For an angle θ in standard position, the coordinates of the point where the terminal side intersects the unit circle are (cosθ,sinθ).

V. Trigonometric Ratios and Angles:

  1. Acute Angles:

    • Trigonometric ratios are primarily defined for acute angles in a right-angled triangle.
  2. Cofunctions:

    • The cofunction identities express the relationships between a trigonometric ratio and its cofunction:
      • sin(π2θ)=cosθ
      • cos(π2θ)=sinθ
      • tan(π2θ)=cotθ
      • cot(π2θ)=tanθ
      • sec(π2θ)=cscθ
      • csc(π2θ)=secθ