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Trigonometric Formula Sheet
Definition of the Trig Functions
Right Triangle Definition Assume that: 0 < θ < π 2 or 0◦^ < θ < 90 ◦
hypotenuse
adjacent
opposite
θ
sin θ = opp hyp csc θ = hyp opp
cos θ = adj hyp sec θ = hyp adj
tan θ = opp adj cot θ = adj opp
Unit Circle Definition Assume θ can be any angle.
x
y
y x
(x, y)
θ
sin θ = y 1 csc θ =
y cos θ = x 1 sec θ =
x tan θ = y x cot θ = x y
Domains of the Trig Functions
sin θ, ∀ θ ∈ (−∞, ∞)
cos θ, ∀ θ ∈ (−∞, ∞)
tan θ, ∀ θ 6 =
n +
π, where n ∈ Z
csc θ, ∀ θ 6 = nπ, where n ∈ Z
sec θ, ∀ θ 6 =
n +
π, where n ∈ Z
cot θ, ∀ θ 6 = nπ, where n ∈ Z
Ranges of the Trig Functions
− 1 ≤ sin θ ≤ 1 − 1 ≤ cos θ ≤ 1 −∞ ≤ tan θ ≤ ∞
csc θ ≥ 1 and csc θ ≤ − 1 sec θ ≥ 1 and sec θ ≤ − 1 −∞ ≤ 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.
sin(ωθ) ⇒ T = 2 π ω cos(ωθ) ⇒ T = 2 π ω tan(ωθ) ⇒ T = π ω
csc(ωθ) ⇒ T = 2 π ω sec(ωθ) ⇒ T = 2 π ω cot(ωθ) ⇒ T = π ω
Identities and Formulas
Tangent and Cotangent Identities
tan θ = sin θ cos θ cot θ = cos θ sin θ
Reciprocal Identities
sin θ =
csc θ
csc θ =
sin θ
cos θ =
sec θ sec θ =
cos θ
tan θ =
cot θ cot θ =
tan θ
Pythagorean Identities
sin^2 θ + cos^2 θ = 1
tan^2 θ + 1 = sec^2 θ
1 + cot^2 θ = csc^2 θ
Even and Odd Formulas
sin(−θ) = − sin θ cos(−θ) = cos θ tan(−θ) = − tan θ
csc(−θ) = − csc θ sec(−θ) = sec θ cot(−θ) = − cot θ
Periodic Formulas If n is an integer
sin(θ + 2πn) = sin θ cos(θ + 2πn) = cos θ tan(θ + πn) = tan θ
csc(θ + 2πn) = csc θ sec(θ + 2πn) = sec θ cot(θ + πn) = cot θ
Double Angle Formulas
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos^2 θ − sin^2 θ = 2 cos^2 θ − 1 = 1 − 2 sin^2 θ
tan(2θ) = 2 tan θ 1 − tan^2 θ
Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then:
π 180 ◦^
t x ⇒ t = πx 180 ◦^ and x = 180 ◦t π
Half Angle Formulas
sin θ = ±
1 − cos(2θ) 2
cos θ = ±
1 + cos(2θ) 2
tan θ = ±
1 − cos(2θ) 1 + cos(2θ) Sum and Difference Formulas
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
tan(α ± β) =
tan α ± tan β 1 ∓ tan α tan β Product to Sum Formulas
sin α sin β =
[cos(α − β) − cos(α + β)]
cos α cos β =
[cos(α − β) + cos(α + β)]
sin α cos β =
[sin(α + β) + sin(α − β)]
cos α sin β =
[sin(α + β) − sin(α − β)]
Sum to Product Formulas
sin α + sin β = 2 sin
α + β 2
cos
α − β 2
sin α − sin β = 2 cos
α + β 2
sin
α − β 2
cos α + cos β = 2 cos
α + β 2
cos
α − β 2
cos α − cos β = −2 sin
α + β 2
sin
α − β 2
Cofunction Formulas
sin
(π 2 − θ
= cos θ
csc
(π 2 − θ
= sec θ
tan
(π 2 − θ
= cot θ
cos
(π 2 − θ
= sin θ
sec
(π 2 − θ
= csc θ
cot
(π 2 − θ
= tan θ
Inverse Trig Functions
Definition
θ = sin−^1 (x) is equivalent to x = sin θ
θ = cos−^1 (x) is equivalent to x = cos θ
θ = tan−^1 (x) is equivalent to x = tan θ
Domain and Range
Function
θ = sin−^1 (x)
θ = cos−^1 (x)
θ = tan−^1 (x)
Domain
− 1 ≤ x ≤ 1
− 1 ≤ x ≤ 1
−∞ ≤ x ≤ ∞
Range
− π 2 ≤ θ ≤ π 2 0 ≤ θ ≤ π
− π 2 < θ < π 2
Inverse Properties These properties hold for x in the domain and θ in the range
sin(sin−^1 (x)) = x
cos(cos−^1 (x)) = x
tan(tan−^1 (x)) = x
sin−^1 (sin(θ)) = θ
cos−^1 (cos(θ)) = θ
tan−^1 (tan(θ)) = θ
Other Notations
sin−^1 (x) = arcsin(x)
cos−^1 (x) = arccos(x)
tan−^1 (x) = arctan(x)
Law of Sines, Cosines, and Tangents
a
b
c
α
β
γ
Law of Sines sin α a
sin β b
sin γ c Law of Cosines
a^2 = b^2 + c^2 − 2 bc cos α
b^2 = a^2 + c^2 − 2 ac cos β
c^2 = a^2 + b^2 − 2 ab cos γ
Law of Tangents
a − b a + b
tan 12 (α − β) tan 12 (α + β)
b − c b + c
tan 12 (β − γ) tan 12 (β + γ)
a − c a + c
tan 12 (α − γ) tan 12 (α + γ)
Complex Numbers
i =
− 1 i^2 = − 1 i^3 = −i i^4 = 1 √ −a = i
a, a ≥ 0
(a + bi) + (c + di) = a + c + (b + d)i
(a + bi) − (c + di) = a − c + (b − d)i
(a + bi)(c + di) = ac − bd + (ad + bc)i
(a + bi)(a − bi) = a^2 + b^2
|a + bi| =
a^2 + b^2 Complex Modulus
(a + bi) = a − bi Complex Conjugate
(a + bi)(a + bi) = |a + bi|^2
DeMoivre’s Theorem
Let z = r(cos θ + i sin θ), and let n be a positive integer. Then: zn^ = rn(cos nθ + i sin nθ).
Example: Let z = 1 − i, find z^6.
Solution: First write z in polar form.
r =
(1)^2 + (−1)^2 =
θ = arg(z) = tan−^1
π 4 Polar Form: z =
cos
π 4
π 4
Applying DeMoivre’s Theorem gives :
z^6 =
cos
π 4
π 4
= 2^3
cos
3 π 2
3 π 2
= 8(0 + i(1))
= 8i
Formulas for the Conic Sections
Circle
StandardF orm : (x − h)^2 + (y − k)^2 = r^2
W here (h, k) = center and r = radius
Ellipse
Standard F orm f or Horizontal M ajor Axis :
(x − h)^2
a^2
(y − k)^2
b^2
Standard F orm f or V ertical M ajor Axis :
(x − h)^2
b^2
(y − k)^2
a^2
Where (h, k)= center
2a=length of major axis
2b=length of minor axis
( 0 < b < a)
Foci can be found by using c^2 = a^2 − b^2
Where c= foci length
More Conic Sections
Hyperbola
Standard F orm f or Horizontal T ransverse Axis :
(x − h)^2
a^2
(y − k)^2
b^2
Standard F orm f or V ertical T ransverse Axis :
(y − k)^2
a^2
(x − h)^2
b^2
Where (h, k)= center
a=distance between center and either vertex
Foci can be found by using b^2 = c^2 − a^2
Where c is the distance between
center and either focus. (b > 0 )
Parabola
Vertical axis: y = a(x − h)^2 + k
Horizontal axis: x = a(y − k)^2 + h
Where (h, k)= vertex
a=scaling factor
x
f (x)
f (x) = tan x
−π 2 π 2
√ 3 3
1
√ 3
− √ 3 3 − 1
−√ 3
−π 3 −π 4 −π 6 0 π 6 π 4 π 3 2 π −π −^56 π −^34 π−^23 π 3 34 π^56 π π
