Trigonometric Identites

Reciprocal and Quotient Identities

  • reciprocal identities
    • cscθ = 1/sinθ
    • secθ = 1/cosθ
    • cotθ = 1/tanθ
  • quotient identities
    • tanθ = sinθ/cosθ
    • cotθ = cosθ/sinθ


Pythagorean Identities

  • sinθ^2 + cosθ^2 = 1
    • sinθ^2 = 1 - cosθ^2
    • cosθ^2 = 1 - sinθ^2
  • tanθ^2 + 1 = secθ^2
    • tanθ^2 = secθ^2 - 1
    • secθ^2 - tanθ^2 = 1
  • 1 + cotθ^2 = cscθ^2
    • cotθ^2 = cscθ^2 - 1
    • cscθ^2 - cotθ^2 = 1


Sum and Difference Identities

  • sin(a + b) = sina*cosb + cosa*sinb
  • sin(a - b) = sina*cosb - cosa*sinb
  • cos(a + b) = cosa*cosb - sina*sinb
  • cos(a - b) = cosa*cosb + sina*sinb
  • tan(a + b) = (tana + tanb)/(1 - tana*tanb)
  • tan(a - b) = (tana - tanb)/(1 + tana*tanb)


Using the Sum and Difference Identities

  • you can find the exact trig ratios of the angles 15°, 75°, 105°, pi/12, 5pi/12, 7pi/12, etc by breaking them up into angles we do know the exact values for (we know 0°, 30°, 45°, 60°, 90°, 0, pi/6, pi/4, pi/3, pi/2)
    • 15° = 45° - 30° (or 60° - 45°)
    • 75° = 45° + 30° (or 60° + 15°)
    • 105° = 90° + 15° (or 60° + 45°)
    • pi/12 = pi/6 + pi/4 (or pi/3 - pi/4)
    • 5pi/12 = pi/6 + pi/4 (or pi/3 + pi/6)
    • 7pi/12 = pi/2 + pi/6 (or pi/3 + pi/4)


Double Angle Identities

  • sin2θ = 2sinθ*cosθ
  • cos2θ = cosθ^2 - sinθ^2
  • cos2θ = 2cosθ^2 - 1
  • cos2θ = 1 - 2sinθ^2
  • tan2θ = (2tanθ)/(1 - tanθ^2)


Finding the Non-Permissible Values of a Trigonometric Expression

  • if sinθ cannot equal 0, θ cannot be n*pi, where n is an integer (every multiple of pi)
  • if cosθ cannot equal 0, θ cannot be (2n + 1)pi/2, where n is an integer (every odd multiple of pi/2)
  • if tanθ cannot equal 0, sinθ and cosθ cannot equal 0, so θ cannot be n*pi or (2n + 1)pi/2

Complete and Continue