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