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Trigonometry Cheat Sheet: Functions, Formulas, and Identities, Exams of Trigonometry

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

2022/2023

Available from 03/26/2024

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TRIGONOMETRY CHEAT SHEET
2
© Paul Dawkins -
https://tutorial.math.lamar.edu
Right triangle definition
Definition of the Trig Functions
Unit Circle Definition
For this definition we assume that
0
<
θ
<
π or 0
<
θ
<
90
.
2
For this definition θ is any angle.
sin(θ)
=
opposite
hypotenus
e
csc(θ) =
hypotenuse
opposite
sin(θ)
=
y
=
ycsc(θ)
=
1
cos(θ)
=
adjacent
sec(θ) = hypotenuse
1y
hypotenus
e
adjacen
t
cos(θ)
=
x
=
x
sec(θ) = 1
tan(θ) =
opposite
adjace
nt
cot(θ) =
adjacent
opposit
e
1
tan(θ) = y
x
x
cot(θ) = x
y
Domain
Facts and Properties
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
+
1
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
=
2π
ω
cos (ω θ)
T
=
2π
ω
csc(θ), θ /= nπ, n
=
0, ±1, ±2, . . . π
sec(θ), θ
n
+
1
2
π, n
=
0, ±1, ±2, . .
.
tan (ω θ)
T
=
ω
2π
cot(θ), θ
nπ, n
=
0, ±1, ±2, . .
.
csc (ω θ)
T
=
ω
2π
sec (ω θ)
T
=
ω
Range
The range is all possible values to get out of the
function.
cot (ω θ)
T
=
π
ω
pf3
pf4
pf5

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Download Trigonometry Cheat Sheet: Functions, Formulas, and Identities and more Exams Trigonometry in PDF only on Docsity!

© Paul Dawkins -

Right triangle definition

Definition of the Trig Functions

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

Facts and Properties

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