Formulas and Identities
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that
!
0 < " < or 0° < " < 90° .
2
Unit circle definition
For this definition " is any angle.
y
( x, y )
hypotenuse
y
opposite
1
"
x
x
"
tan " + 1 = sec "
adjacent
opposite
hypotenuse
adjacent
cos " =
hypotenuse
opposite
tan " =
adjacent
sin " =
2
hypotenuse
opposite
hypotenuse
sec" =
adjacent
adjacent
cot " =
opposite
csc " =
y
=y
1
x
cos " = = x
1
y
tan " =
x
sin " =
1
y
1
sec " =
x
x
cot " =
y
csc " =
Facts and Properties
Domain
The domain is all the values of " that
can be plugged into the function.
sin " ,
cos " ,
tan " ,
csc " ,
sec " ,
cot " ,
Tangent and Cotangent Identities
sin "
cos "
tan " =
cot " =
cos "
sin "
Reciprocal Identities
1
1
csc " =
sin " =
sin "
csc "
1
1
sec " =
cos " =
cos "
sec "
1
1
cot " =
tan " =
tan "
cot "
Pythagorean Identities
sin 2 " + cos 2 " = 1
" can be any angle
" can be any angle
1"
!
" # $ n + % ! , n = 0, ± 1, ± 2,…
2'
&
" # n ! , n = 0, ± 1, ± 2,…
1"
!
" # $ n + % ! , n = 0, ± 1, ± 2,…
2'
&
" # n ! , n = 0, ± 1, ± 2,…
Range
The range is all possible values to get
out of the function.
csc " * 1 and csc " ) (1
(1 ) sin " ) 1
sec " * 1 and sec " ) (1
(1 ) cos " ) 1
(+ < tan " < +
(+ < cot " < +
Period
The period of a function is the number,
T, such that f (" + T ) = f (" ) . So, if #
is a fixed number and " is any angle we
have the following periods.
sin ( #" ) ,
T
cos (#" ) ,
T
tan (#" ) ,
T
csc (#" ) ,
T
sec (#" ) ,
T
cot (#" ) ,
T
2!
=
#
2!
=
#
!
=
#
2!
=
#
2!
=
#
!
=
#
© 2005 Paul Dawkins
2
1 + cot 2 " = csc 2 "
Even/Odd Formulas
sin ( (" ) = ( sin "
csc ( (" ) = ( csc "
cos ( (" ) = cos "
sec ( (" ) = sec "
tan ( (" ) = ( tan "
cot ( (" ) = ( cot "
Periodic Formulas
If n is an integer.
sin (" + 2! n ) = sin "
csc (" + 2! n ) = csc "
cos (" + 2! n ) = cos " sec (" + 2! n ) = sec "
tan (" + ! n ) = tan "
cot (" + ! n ) = cot "
Double Angle Formulas
sin ( 2" ) = 2sin " cos "
cos ( 2" ) = cos 2 " ( sin 2 "
= 2 cos 2 " ( 1
= 1 ( 2sin 2 "
tan ( 2" ) =
2 tan "
1 ( tan 2 "
Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then
!
t
!x
180t
=
- t=
and x =
180 x
180
!
Half Angle Formulas
1
sin 2 " = (1 ( cos ( 2" ) )
2
1
cos 2 " = (1 + cos ( 2" ) )
2
1 ( cos ( 2" )
2
tan " =
1 + cos ( 2" )
Sum and Difference Formulas
sin ($ ± % ) = sin $ cos % ± cos $ sin %
cos ($ ± % ) = cos $ cos % ! sin $ sin %
tan $ ± tan %
1 ! tan $ tan %
Product to Sum Formulas
1
sin $ sin % = .0 cos ($ ( % ) ( cos ($ + % ) /1
2
1
cos $ cos % = .0 cos ($ ( % ) + cos ($ + % ) /1
2
1
sin $ cos % = .0sin ($ + % ) + sin ($ ( % ) /1
2
1
cos $ sin % = .0sin ($ + % ) ( sin ($ ( % ) /1
2
Sum to Product Formulas
!$ + % "
!$ ( % "
sin $ + sin % = 2 sin $
% cos $
%
2
&
'
& 2 '
!$ + % " !$ ( % "
sin $ ( sin % = 2 cos $
% sin $
%
& 2 ' & 2 '
tan ($ ± % ) =
!$ + % "
!$ ( % "
cos $ + cos % = 2 cos $
% cos $
%
2
&
'
& 2 '
!$ + % " !$ ( % "
cos $ ( cos % = (2 sin $
% sin $
%
& 2 ' & 2 '
Cofunction Formulas
!!
"
sin $ ( " % = cos "
&2
'
!!
"
csc $ ( " % = sec "
&2
'
!!
"
cos $ ( " % = sin "
&2
'
!!
"
sec $ ( " % = csc "
&2
'
!!
"
tan $ ( " % = cot "
2
&
'
!!
"
cot $ ( " % = tan "
2
&
'
© 2005 Paul Dawkins
Unit Circle
y
!
3 1"
$( , %
& 2 2'
3!
4
( 0,1)
!
2
! 1 3"
$( , %
& 2 2 '
!
2 2"
,
$(
%
& 2 2 '
Inverse Trig Functions
2!
3
!
3
90°
120°
45°
30°
! 180°
y = cos ( 1 x is equivalent to x = cos y
sin ( sin (1 ( x ) ) = x
sin (1 ( sin (" ) ) = "
tan ( tan ( 1 ( x ) ) = x
tan (1 ( tan (" ) ) = "
y = tan x is equivalent to x = tan y
! 2 2"
,
$$
%%
& 2 2 '
!
6
Domain and Range
Function
Domain
! 3 1"
$$ 2 , 2 %%
&
'
y = sin (1 x
(1 ) x ) 1
y = cos x
(1 ) x ) 1
y = tan (1 x
(+ < x < +
(1
150°
( (1,0 )
Inverse Properties
cos ( cos (1 ( x ) ) = x
cos (1 ( cos (" ) ) = "
(1
!
4
60°
135°
5!
6
!1 3"
$$ , %%
&2 2 '
Definition
y = sin ( 1 x is equivalent to x = sin y
0°
0
360°
2!
(1,0 )
!
3 1"
$ ( ,( %
2'
& 2
!
2
2"
,(
$(
%
2 '
& 2
210°
330°
225°
5!
4
4!
3
240°
! 1
3"
$ ( ,(
%
& 2 2 '
270°
3!
2
315°
7!
300°
4
5!
3
( 0,(1)
11!
6
!1
3"
$ ,(
%
&2 2 '
For any ordered pair on the unit circle ( x, y ) : cos " = x and sin " = y
a
&
Law of Sines
sin $ sin % sin &
=
=
a
b
c
Law of Tangents
a ( b tan 12 ($ ( % )
=
a + b tan 12 ($ + % )
Law of Cosines
a 2 = b 2 + c 2 ( 2bc cos $
b ( c tan 12 ( % ( & )
=
b + c tan 12 ( % + & )
c = a + b ( 2ab cos &
2
! 5! " 1
cos $
%=
& 3 ' 2
%
b
b 2 = a 2 + c 2 ( 2ac cos %
Example
tan ( 1 x = arctan x
$
! 3 1"
$ ,( %
& 2 2'
! 2
2"
,(
$
%
2 '
& 2
cos ( 1 x = arccos x
Law of Sines, Cosines and Tangents
x
c
7!
6
Alternate Notation
sin (1 x = arcsin x
Range
!
!
( ) y)
2
2
0) y )!
!
!
( < y<
2
2
2
2
a ( c tan 12 ($ ( & )
=
a + c tan 12 ($ + & )
Mollweide’s Formula
a + b cos 12 ($ ( % )
=
c
sin 12 &
3
! 5! "
sin $
%=(
2
& 3 '
© 2005 Paul Dawkins
© 2005 Paul Dawkins