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Trigonometric Formula SheetDefinition of the Trig Functions
Right Triangle Definition Unit Circle Definition
Assume that: Assume θ can be any angle.
0 < θ < π2 or 0◦ < θ < 90◦
y
(x, y)
hypotenuse 1
y
opposite θ
x
x
θ
adjacent
opp hyp
sin θ = csc θ = y 1
hyp opp sin θ = csc θ =
1 y
adj hyp x 1
cos θ = sec θ = cos θ = sec θ =
hyp adj 1 x
opp adj y x
tan θ = cot θ = tan θ = cot θ =
adj opp x y
Domains of the Trig Functions
sin θ, ∀ θ ∈ (−∞, ∞) csc θ, ∀ θ 6= nπ, where n ∈ Z
1
cos θ, ∀ θ ∈ (−∞, ∞) sec θ, ∀ θ 6= n + π, where n ∈ Z
2
1
tan θ, ∀ θ 6= n + π, where n ∈ Z cot θ, ∀ θ 6= nπ, where n ∈ Z
2
Ranges of the Trig Functions
−1 ≤ sin θ ≤ 1 csc θ ≥ 1 and csc θ ≤ −1
−1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1
−∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞
Periods of the Trig Functions
The period of a function is the number, T, such that f (θ +T ) = f (θ ) .
So, if ω is a fixed number and θ is any angle we have the following periods.
2π 2π
sin(ωθ) ⇒ T = csc(ωθ) ⇒ T =
ω ω
2π 2π
cos(ωθ) ⇒ T = sec(ωθ) ⇒ T =
ω ω
π π
tan(ωθ) ⇒ T = cot(ωθ) ⇒ T =
ω ω
1
, Identities and Formulas
Tangent and Cotangent Identities Half Angle Formulas
r
sin θ cos θ 1 − cos(2θ)
tan θ = cot θ = sin θ = ±
cos θ sin θ 2
r
Reciprocal Identities 1 + cos(2θ)
cos θ = ±
1 1 2
sin θ = csc θ = s
csc θ sin θ 1 − cos(2θ)
1 1 tan θ = ±
cos θ = sec θ = 1 + cos(2θ)
sec θ cos θ
Sum and Difference Formulas
1 1
tan θ = cot θ =
cot θ tan θ sin(α ± β) = sin α cos β ± cos α sin β
Pythagorean Identities cos(α ± β) = cos α cos β ∓ sin α sin β
2 2
sin θ + cos θ = 1
tan α ± tan β
tan2 θ + 1 = sec2 θ tan(α ± β) =
1 ∓ tan α tan β
1 + cot2 θ = csc2 θ
Product to Sum Formulas
Even and Odd Formulas
1
sin α sin β = [cos(α − β) − cos(α + β)]
sin(−θ) = − sin θ csc(−θ) = − csc θ 2
cos(−θ) = cos θ sec(−θ) = sec θ 1
cos α cos β = [cos(α − β) + cos(α + β)]
tan(−θ) = − tan θ cot(−θ) = − cot θ 2
1
Periodic Formulas sin α cos β = [sin(α + β) + sin(α − β)]
2
If n is an integer 1
cos α sin β = [sin(α + β) − sin(α − β)]
sin(θ + 2πn) = sin θ csc(θ + 2πn) = csc θ 2
cos(θ + 2πn) = cos θ sec(θ + 2πn) = sec θ Sum to Product Formulas
tan(θ + πn) = tan θ cot(θ + πn) = cot θ
α+β α−β
Double Angle Formulas sin α + sin β = 2 sin cos
2 2
α+β α−β
sin(2θ) = 2 sin θ cos θ sin α − sin β = 2 cos sin
2 2
cos(2θ) = cos2 θ − sin2 θ α+β α−β
cos α + cos β = 2 cos cos
= 2 cos2 θ − 1 2 2
= 1 − 2 sin2 θ α+β α−β
cos α − cos β = −2 sin sin
2 2
2 tan θ
tan(2θ) = Cofunction Formulas
1 − tan2 θ
π π
Degrees to Radians Formulas sin − θ = cos θ cos − θ = sin θ
If x is an angle in degrees and t is an angle in 2 2
π π
radians then: csc − θ = sec θ sec − θ = csc θ
2 2
π t πx 180◦ t π π
= ⇒ t = and x = tan − θ = cot θ cot − θ = tan θ
180◦ x 180◦ π 2 2
2
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