Trig cheat sheet

Trigonometric formulas

1. © 2005 Paul Dawkins
Trig Cheat Sheet

Definition of the Trig Functions
Right triangle definition
For this definition we assume that
0
2
p
q<< or 090q°<<° .


opposite
sin
hypotenuse
q=
hypotenuse
csc
opposite
q=
adjacent
cos
hypotenuse
q=
hypotenuse
sec
adjacent
q=
opposite
tan
adjacent
q=
adjacent
cot
opposite
q=


Unit circle definition
For this definition q is any angle.

sin
1
y
yq==
1
csc
y
q=
cos
1
x
xq==
1
sec
x
q=
tan
y
x
q= cot
x
y
q=
Facts and Properties
Domain
The domain is all the values of q that
can be plugged into the function.

sinq , q can be any angle
cosq, q can be any angle
tanq,
1
,0,1,2,
2
nnqp
æö
¹+=±±
ç÷
èø
K
cscq, ,0,1,2,nnqp¹=±± K
secq,
1
,0,1,2,
2
nnqp
æö
¹+=±±
ç÷
èø
K
cotq, ,0,1,2,nnqp¹=±± K

Range
The range is all possible values to get
out of the function.
1sin1q-££ csc1andcsc1qq³£-
1cos1q-££ sec1andsec1qq³£-
tanq-¥<<¥ cotq-¥<<¥

Period
The period of a function is the number,
T, such that ( )()fTfqq+= . So, if w
is a fixed number and q is any angle we
have the following periods.

()sinwq®
2
T
p
w
=
()coswq®
2
T
p
w
=
()tanwq® T
p
w
=
()cscwq®
2
T
p
w
=
()secwq®
2
T
p
w
=
()cotwq® T
p
w
=
q
adjacent
opposite
hypotenuse
x
y
(),xy
q
x
y
1

2. © 2005 Paul Dawkins
Formulas and Identities
Tangent and Cotangent Identities
sincos
tancot
cossin
qq
qq
qq
==
Reciprocal Identities
11
cscsin
sincsc
11
seccos
cossec
11
cottan
tancot
qq
qq
qq
qq
qq
qq
==
==
==

Pythagorean Identities
22
22
22
sincos1
tan1sec
1cotcsc
qq
qq
qq
+=
+=
+=

Even/Odd Formulas
() ()
()()
()()
sinsincsccsc
coscossecsec
tantancotcot
qqqq
qqqq
qqqq
-=--=-
-=-=
-=--=-

Periodic Formulas
If n is an integer.
( ) ( )
()()
()()
sin2sincsc2csc
cos2cossec2sec
tantancotcot
nn
nn
nn
qpqqpq
qpqqpq
qpqqpq
+=+=
+=+=
+=+=
Double Angle Formulas
()
()
()
22
2
2
2
sin22sincos
cos2cossin
2cos1
12sin
2tan
tan2
1tan
qqq
qqq
q
q
q
q
q
=
=-
=-
=-
=
-

Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then
180
and
180180
txt
tx
x
pp
p
=Þ==
Half Angle Formulas
()()
()()
()
()
2
2
2
1
sin1cos2
2
1
cos1cos2
2
1cos2
tan
1cos2
qq
qq
q
q
q
=-
=+
-
=
+

Sum and Difference Formulas
( )
()
()
sinsincoscossin
coscoscossinsin
tantan
tan
1tantan
ababab
ababab
ab
ab
ab
±=±
±=
±
±=
m
m

Product to Sum Formulas
()()
()()
()()
()()
1
sinsincoscos
2
1
coscoscoscos
2
1
sincossinsin
2
1
cossinsinsin
2
ababab
ababab
ababab
ababab
=--+éù
ëû
=-++éù
ëû
=++-éù
ëû
=+--éù
ëû
Sum to Product Formulas
sinsin2sincos
22
sinsin2cossin
22
coscos2coscos
22
coscos2sinsin
22
abab
ab
abab
ab
abab
ab
abab
ab
+-æöæö
+=
ç÷ç÷
èøèø
+-æöæö
-=
ç÷ç÷
èøèø
+-æöæö
+=
ç÷ç÷
èøèø
+-æöæö
-=-
ç÷ç÷
èøèø
Cofunction Formulas
sincoscossin
22
cscsecseccsc
22
tancotcottan
22
pp
qqqq
pp
qqqq
pp
qqqq
æöæö
-=-=
ç÷ç÷
èøèø
æöæö
-=-=
ç÷ç÷
èøèø
æöæö
-=-=
ç÷ç÷
èøèø

3. © 2005 Paul Dawkins



Unit Circle



For any ordered pair on the unit circle (),xy : cos xq= and sinyq=

Example
5153
cossin
3232
ppæöæö
==-
ç÷ç÷
èøèø



3
p

4
p

6
p

22
,
22
æö
ç÷
ç÷
èø

31
,
22
æö
ç÷
ç÷
èø

13
,
22
æö
ç÷
ç÷
èø

60°

45°
30°
2
3
p

3
4
p

5
6
p

7
6
p

5
4
p

4
3
p

11
6
p

7
4
p

5
3
p

2
p

p
3
2
p

0
2p

13
,
22
æö
-ç÷
èø

22
,
22
æö
-ç÷
èø

31
,
22
æö
-ç÷
èø

31
,
22
æö
--ç÷
èø

22
,
22
æö
--ç÷
èø
13
,
22
æö
--ç÷
èø
31
,
22
æö
-ç÷
èø

22
,
22
æö
-ç÷
èø

13
,
22
æö
-ç÷
èø

()0,1
()0,1-
()1,0-
90°
120°
135°
150°
180°
210°
225°
240°
270°

300°
315°
330°
360°

0°
x
()1,0
y

4. © 2005 Paul Dawkins




Inverse Trig Functions
Definition
1
1
1
sin is equivalent to sin
cos is equivalent to cos
tan is equivalent to tan
yxxy
yxxy
yxxy
-
-
-
==
==
==


Domain and Range
Function Domain Range
1
sinyx
-
= 11x-££
22
y
pp
-££
1
cosyx
-
= 11x-££ 0yp££
1
tanyx
-
= x-¥<<¥
22
y
pp
-<<

Inverse Properties
()( ) ()( )
()() ()()
()() ()()
11
11
11
coscoscoscos
sinsinsinsin
tantantantan
xx
xx
xx
qq
qq
qq
--
--
--
==
==
==


Alternate Notation
1
1
1
sinarcsin
cosarccos
tanarctan
xx
xx
xx
-
-
-
=
=
=

Law of Sines, Cosines and Tangents


Law of Sines
sinsinsin
abc
abg
==
Law of Cosines
222
222
222
2cos
2cos
2cos
abcbc
bacac
cabab
a
b
g
=+-
=+-
=+-

Mollweide’s Formula
( )
1
2
1
2
cos
sin
ab
c
ab
g
-+
=
Law of Tangents
( )
()
()
()
()
()
1
2
1
2
1
2
1
2
1
2
1
2
tan
tan
tan
tan
tan
tan
ab
ab
bc
bc
ac
ac
ab
ab
bg
bg
ag
ag
--
=
++
--
=
++
--
=
++

c a
b
a
b
g