Trigonometric Functions Overview

Key Concepts of Trigonometric Functions

• Definitions via SOH CAH TOA for Right Triangles:

  • Sine (sin) = Opposite/Hypotenuse
  • Cosine (cos) = Adjacent/Hypotenuse
  • Tangent (tan) = Opposite/Adjacent

• Trigonometric Functions on Unit Circle:

  • Defined for all angles, not limited to 0-90 degrees.
  • For a standard position angle θ:
  • Coordinates of the intersection point where the terminal side touches the unit circle: (cos(θ), sin(θ))
  • tan(θ) = sin(θ)/cos(θ), undefined when cos(θ) = 0.

• Function Outputs:

  • Sine = y-coordinate (sin(θ) = y)
  • Cosine = x-coordinate (cos(θ) = x)
  • Tangent = y/x, undefined when x = 0.
  • Csc (cosecant) = 1/y, undefined when y = 0.
  • Sec (secant) = 1/x, undefined when x = 0.
  • Cot (cotangent) = x/y, undefined when y = 0.

• Radians and Degree Conversion:

  • Conversion practice: angle measure in radians for standard-position angles.

• Examples of exact values from the unit circle:

  • sin(0) = 0, cos(π) = -1,
  • sin(90°) = 1, sin(-π/2) = -1,
  • cos(-π/2) = 0, cos(π/6) = √3/2,
  • sin(π/6) = 1/2, tan(π/6) = 1/√3,
  • tan(3π/4) = -1, sin(3π) = 0,
  • sec(5π/6) = -2/√3, cos(-3π) = -1,
  • tan(3π/2) undefined, csc(390°) = √2,
  • cot(-π/4) = -1.

• True/False Statements:

  • Notation like cos(0) = 1 signifies the cosine of 0 degrees, which equals 1, not a multiplication.