Definition of the Trig Functions
Right Triangle Definition
For this definition assume:
0˂θ˂
!
!
Formulas and Identities
Unit Circle Definition
For this definition θ is
or 0° ˂ θ ˂ 90°
Tangent and Cotangent
tanθ =
any angle
!"#!
cotθ =
!"#!
!"#!
!"#!
Reciprocal Identities
cscθ =
secθ =
cotθ =
sinθ =
cosθ =
tanθ =
!""!#$%&
!"#$%&'()&
!"#!$%&'
!"#$%&'()&
!""!#$%&
!"#!$%&'
cscθ =
secθ =
!"#$%&'()&
!""!#$%&
!"#$%&'()&
!"#!$%&'
!"#!$%&'
cotθ = !""!#$%&
!
sinθ = ! = y
!
cosθ = ! = x
!
tanθ = !
!
cscθ = !
!
secθ = !
!
cotθ = !
Graphing and Properties
Domain
All of the values of θ that can be
plugged into the function.
sinθ, θ can be any angle
cosθ, θ can be any angle
Period
Period is a number T, such that
f (θ + T) =f (θ). If ω is a fixed
number and θ is any angle we
have the following periods.
tanθ, θ ≠ (n+1/2)π, n = 0, ±1, ±2,…
sin(ωθ) → T =
cscθ, θ ≠ nπ, n = 0, ±1, ±2,…
cos(ωθ) → T =
secθ, θ ≠ (n+1/2)π, n = 0, ±1, ±2,…
tan(ωθ) → T =
!"
!
!"
!
!
!
csc(ωθ) → T =
sec(ωθ) → T =
cot(ωθ) → T =
!
sinθ =
!"#!
!
!"#!
!
!"#!
cosθ =
tanθ =
Half Angle Formulas
!
sin
!
tan
!
!"!!
!
!"#!
!
!"#!
=±
!
=±
!
!!!"#!
!
cos
!
!!!"#!
!!!"#!
=
!!!"#!
!"#!
!
=±
!!!"#!
!
!"#!
=
!!!"#!
!
!
sin2θ = (1-cos(2θ)); cos2θ = (1+cos(2θ))
2
tan θ =
!
!!!"# (!")
!
!!!"# (!")
Sum and Difference Formulas
Pythagorean Identities
sin2θ + cos2θ = 1
sin(α ± β) = sinαcosβ±cosαsinβ
cos(α ± β) = cosαcosβ∓sinαsinβ
tan2θ + 1 = sec2θ
tan(α ± β) =
1 + cot2θ = csc2θ
Product to Sum Formulas
Even/Odd Formulas
sinαsinβ = [cos(α - β) - cos(α + β)]
sin(-θ) = -sinθ
csc(-θ) = -cscθ
cosαcosβ = [cos(α - β) + cos(α + β)]
cos(-θ) = cosθ
sec(-θ) = secθ
sinαcosβ = [sin(α + β) + sin(α - β)]
tan(-θ) = -tanθ
cot(-θ) = -cotθ
cosαsinβ = [sin(α + β) - sin(α - β)]
!"#! ± !"#!
!∓!"#!!"#!
!
!
!
!
!
!
!
!
Periodic Formulas
Sum to Product Formulas
If n is an integer
sinα + sinβ = 2sin
!
!"
sin(θ + 2πn) = sinθ
sinα - sinβ = 2cos
!
!
csc(θ + 2πn) = cscθ
cosα + cosβ = 2cos
!
cos(θ + 2πn) = cosθ
cosα - cosβ = -2sin
sec(θ + 2πn) = secθ
Cofunction Formulas
tan(θ + πn) = tanθ
sin
cot(θ + πn) = cotθ
csc
Double Angle Formulas
tan
sin(2θ) = 2sinθcosθ
cos(2θ) = cos2θ – sin2θ
= 2cos2θ – 1
Degrees to Radians Formulas
If x is an angle in degrees and θ is an
angle in radians then:
!"
cotθ, θ ≠ nπ, n = 0, ±1, ±2,…
if -1 ˂ ω ˂ 1 then shrunk horizontally
Range
if ω ˂ -1 or 1 ˂ ω then stretched
All possible values that come out
Amplitude
of the function.
y = Asinθ or y = Acosθ
-1 ≤ sinθ ≤ 1
cscθ ≥ 1 and cscθ ≤ -1
if -1 ˂ A ˂ 1 then shrunk vertically
-1 ≤ cosθ ≤ 1 secθ ≥ 1 and secθ ≤ -1
if A ˂ -1 or 1 ˂ A then stretched
-∞ ˂ tanθ ˂ ∞ -∞ ˂ cotθ ˂ ∞
Distance from min to max = 2A
Shifts
y = sin(θ – c) (applies to all trig func.)
y = sinθ + d (applies to all trig func.)
shift right by c units
shift up by d units
y = sin(θ + c) (applies to all trig func.)
y = sinθ - d (applies to all trig func.)
shift left by c units
shift down by d units
= 1 - 2sin2θ
tan(2θ) =
!"#$!
!!!"#! !
!
!
!
!
!
!
!
!"#°
!! !
!
!! !
cos
sin
!
!! !
!
!! !
!
− θ = cosθ
cos
− θ = secθ
sec
− θ = cotθ
cot
=
!
!
→θ=
!!
!"#°
!! !
!
!! !
cos
sin
!
!
!
!
!
!
!
!! !
!
!! !
!
− θ = sinθ
− θ = cscθ
− θ = tanθ
and x =
!"#°!
!
Compiled with Paul Dawkins’ 2005 Trig Cheat Sheet