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A comprehensive cheat sheet on trigonometry functions, formulas, and identities. It covers the definitions of sine, cosine, tangent, and cotangent, their domains and ranges, periodicity, and various formulas and identities. It also includes conversions between degrees and radians.
Typology: Exams
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© Paul Dawkins -
Right triangle definition
Unit Circle Definition
For this definition we assume that
0 < θ <
π
or 0
◦
< θ < 90
◦
For this definition θ is any angle.
sin( θ ) =
opposite
hypotenus
e
csc( θ ) = hypotenuse
opposite
sin( θ ) =
y
= y csc( θ ) =
cos ( θ ) =
adjacent
sec ( θ ) =
hypotenuse
1 y
hypotenus
e
adjacen
t
cos( θ ) =
x
= x sec( θ ) =
tan( θ ) = opposite
adjace
nt
cot adjacent( θ ) =
opposit
e
tan( θ ) =
y
x
x
cot( θ ) =
x
y
Domain
Period
The domain is all the values of θ that
can be plugged into the function.
sin( θ ), θ can be any
angle cos( θ ), θ can be
any angle
tan( θ ), θ n +
π, n = 0 , ± 1 , ± 2 ,...
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 ( ω θ ) T =
2 π
ω
cos ( ω θ ) T =
2 π
ω
csc( θ ), θ /= nπ, n = 0 , ± 1 , ± 2 ,...
π
sec( θ ), θ n +
π, n = 0 , ± 1 , ± 2 ,..
tan ( ω θ ) → T =
ω
2 π
cot( θ ), θ nπ, n = 0 , ± 1 , ± 2 ,..
csc ( ω θ ) → T =
ω
2 π
sec ( ω θ ) → T =
ω
Range
The range is all possible values to get out of the
function.
cot ( ω θ ) T =
π
ω
© Paul Dawkins -
− 1 ≤ sin( θ ) ≤ 1 − 1 ≤ cos( θ ) ≤ 1
−∞ < tan( θ ) < ∞ −∞ < cot( θ ) < ∞
sec( θ ) ≥ 1 and sec( θ ) ≤ − 1 csc( θ ) ≥ 1 and csc( θ ) ≤ − 1
tan
2
( θ )
© Paul Dawkins -
x
⇒ t =
and x =
π
cos( α ) + cos( β ) = 2 cos
α + β
cos
α −
β
Double Angle Formulas
sin(2 θ ) = 2 sin( θ ) cos( θ )
cos(2 θ ) = cos
2
( θ ) −
sin
2
( θ )
= 2 cos
2
( θ ) − 1
sin
2
( θ )
tan (2 θ ) =
tan( θ )
cos( α )−cos( β ) = − 2 sin
α + β
sin
α −
β
Cofunction Formulas
sin
π
− θ = cos( θ ) cos
π
− θ = sin( θ )
csc
π
− θ = sec( θ ) sec
π
− θ = csc( θ )
tan
π
− θ = cot( θ ) cot
π
− θ = tan( θ )
© Paul Dawkins -
For any ordered pair on the unit circle ( x, y ) : cos( θ ) = x and sin( θ ) = y
Example
cos
5 π
sin
5 π