Trig Cheat Sheet
Formulas and Identities
Definition of the Trig Functions
Right triangle definition
Tangent and Cotangent Identities
Half Angle Formulas
sinB
tan 8 =-cos8
sin?
For this definition we assume that
Unit circle defmition
Reciprocal
o < B <!!.-
For this definition 8 is any angle.
cscB=-- 1
sin8
1
secB=-cosB
1
cot8=-tanB
or 0° < B < 90° .
2
.•.y
~I
hypotenuse
~~
opposite
L
. B
opposite
sm =-"--'--hypotenuse
adjacent
cos8
hypotenuse
opposite
tan B =--adjacent
adjacent
sin8=~=
y
1
cosB=~=x
csc8=secB=-
1
tan8=~
The domain is all the values of B that
can be plugged into the function.
8"" (n+±}r,
csc8,
8""n7[,
sec8,
8"" (n+±)7[,
cotB,
8""n7[,
±2, ...
n = 0,±1,±2, ...
n=O,±I,
±2, ...
Range
The range is all possible values to get
out of the function.
-15 sin B :S1
csc 8 2 1 and csc 8 :S-1
-1:S cos8:S 1 sec8 21 andsec8:S-1
-00 :Stan 8 :S00
-00 :Scot B :S00
csc' B
1
sin( -8) = -sinB
csc( -B) = - cscB
x
cos (-8) = cos B
sec( -B) = sec8
tan ( -B) = -tanB
cot (-B) = - cote
y
csc( B + 27[n) = cscB
The period of a function is the number,
T, such that f (B + T) = f (B). So, if W
cos( B + 27[n) = cosB
sec( 8+ 27[n) = secB
tan( B +7[n) = tan 8
cot( B+ 7[n) = cote
is a fixed number and B is any angle we
have the following periods.
Double Angle Formulas
sin(w8)
-7
T= 27[
W
cos(wB)
-7
tan(wB)
-7
csc( w B)
-7
sec(wB)
-7
cot( w B)
-7
T= 27[
w
T=~
w
T= 27[
w
T= 27[
w
T=~
©
tan2B
l-cos(28)
l+cos(28)
Sum and Difference Formulas
sin (a ± fJ) = sin a cos fJ ± cosa sin fJ
cos( a ± fJ)
= cosa cos fJ =t=sinasin fJ
tan ( a ± fJ)
tana±tanfJ
= -----'--
l=t=tanatanfJ
cosa cas 13= ~[cos( a- fJ)+ cos( a + fJ)]
2
= ~[sin( a + f3)+sin (a - 13)]
sina cosf3
2
cosa sin 13= ~[sin (a + 13) -sin( a - 13)]
sin( B + 27[n) = sinB
Period
cos28 = ~(1+ cos(2B))
2
sin a sin fJ = ~[cos (a - 13) - cas (a + 13)]
2
Periodic Formulas
If n is an integer.
n = 0,±1,±2, ...
n=O,±I,
=
Even/Odd Formulas
cotB=~
x
1 + cor' B
e = ~(Icos (28))
2
Product to Sum Formulas
1
y
Facts and Properties
tan 8 ,
Identities
sin ' B + cos" 8 = 1
Domain
8 can be any angle
8 can be any angle
. B =--1
Sin
csc8
1
cosB=-secB
1
tan8=-cote
tan! B + 1 = see! B
hypotenuse
csc B =-~-opposite
hypotenuse
sec B =-~-adjacent
adjacent
cot B = ---''--opposite
sin8,
cos8,
sinB
Identities
Pythagorean
.~~
e cosB
cot =--
2
Sum to Product Formulas
.
. fJ = 2·sin (a+
sma+sm
-2- fJ) cas (a--2- fJ)
f3
.
. 13 = 2 cas (a+
sma-sm
-2- ).
fJ
sin (a-2-)
sin (28) = 2sin8cos8
cos(2B) = cos' B- sin" 8
cosa+ cas fJ = 2COS(a ~ f3)cos(
a; fJ)
= 2cos2 B-1
= 1- 2sin2 B
tan (2B) = 2 tan 8
I-tan2B
Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then
7[
t
7[X
180t
t=and X=-180 x
7[
180
cosa -cosfJ
= -2sin( a~ f3)sin( a; 13)
Cofunction
Formulas
Sin(
csc(
~-B)
~-B)
= cosB
COS(~-B )=sinB
= secB
sec(
tan(~-B)=CotB
~-B)
cot(~-B)=
= cscB
tanB
w
2005 Paul Dawkins
©
2005 Paul Dawkins