# trig cheat sheet reduced

```Formulas and Identities
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that
p
0 &lt; q &lt; or 0&deg; &lt; q &lt; 90&deg; .
2
Unit circle definition
For this definition q is any angle.
y
( x, y )
hypotenuse
y
opposite
1
q
x
x
q
tan 2 q + 1 = sec 2 q
opposite
hypotenuse
cos q =
hypotenuse
opposite
tan q =
sin q =
hypotenuse
opposite
hypotenuse
sec q =
cot q =
opposite
csc q =
y
sin q = = y
1
x
cos q = = x
1
y
tan q =
x
1
csc q =
y
1
sec q =
x
x
cot q =
y
Facts and Properties
Domain
The domain is all the values of q that
can be plugged into the function.
sin q ,
cos q ,
tan q ,
q can be any angle
q can be any angle
1&ouml;
&aelig;
q &sup1; &ccedil; n + &divide; p , n = 0, &plusmn; 1, &plusmn; 2, K
2&oslash;
&egrave;
q &sup1; n p , n = 0, &plusmn; 1, &plusmn; 2, K
Period
The period of a function is the number,
T, such that f (q + T ) = f (q ) . So, if w
is a fixed number and q is any angle we
have the following periods.
2p
w
2p
=
w
p
=
w
2p
=
w
2p
=
w
p
=
w
sin ( wq ) &reg;
T=
1&ouml;
&aelig;
q &sup1; &ccedil; n + &divide; p , n = 0, &plusmn; 1, &plusmn; 2, K
2&oslash;
&egrave;
q &sup1; n p , n = 0, &plusmn; 1, &plusmn; 2, K
cos (wq ) &reg;
T
tan (wq ) &reg;
T
Range
The range is all possible values to get
out of the function.
csc q &sup3; 1 and csc q &pound; -1
-1 &pound; sin q &pound; 1
sec q &sup3; 1 and sec q &pound; -1
-1 &pound; cos q &pound; 1
-&yen; &lt; tan q &lt; &yen;
-&yen; &lt; cot q &lt; &yen;
csc (wq ) &reg;
T
sec ( wq ) &reg;
T
cot ( wq ) &reg;
T
csc q ,
sec q ,
cot q ,
Tangent and Cotangent Identities
sin q
cos q
tan q =
cot q =
cos q
sin q
Reciprocal Identities
1
1
csc q =
sin q =
sin q
csc q
1
1
secq =
cos q =
cos q
sec q
1
1
cot q =
tan q =
tan q
cot q
Pythagorean Identities
sin 2 q + cos 2 q = 1
&copy; 2005 Paul Dawkins
1 + cot 2 q = csc 2 q
Even/Odd Formulas
sin ( -q ) = - sin q
csc ( -q ) = - csc q
cos ( -q ) = cos q
sec ( -q ) = sec q
tan ( -q ) = - tan q
cot ( -q ) = - cot q
Periodic Formulas
If n is an integer.
sin (q + 2p n ) = sin q
csc (q + 2p n ) = csc q
cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q
tan (q + p n ) = tan q
cot (q + p n ) = cot q
Double Angle Formulas
sin ( 2q ) = 2sin q cos q
cos ( 2q ) = cos 2 q - sin 2 q
= 2 cos 2 q - 1
= 1 - 2 sin 2 q
2 tan q
tan ( 2q ) =
1 - tan 2 q
If x is an angle in degrees and t is an
p
t
px
180t
=
&THORN; t=
and x =
180 x
180
p
Half Angle Formulas
(alternate form)
q
1 - cos q
1
sin = &plusmn;
sin 2 q = (1 - cos ( 2q ) )
2
2
2
cos
q
1 + cos q
=&plusmn;
2
2
cos 2 q =
1
(1 + cos ( 2q ) )
2
1 - cos ( 2q )
q
1 - cos q
=&plusmn;
tan 2 q =
2
1 + cos q
1 + cos ( 2q )
Sum and Difference Formulas
sin (a &plusmn; b ) = sin a cos b &plusmn; cos a sin b
tan
cos (a &plusmn; b ) = cos a cos b m sin a sin b
tan a &plusmn; tan b
1 m tan a tan b
Product to Sum Formulas
1
sin a sin b = &eacute;&euml;cos (a - b ) - cos (a + b ) &ugrave;&ucirc;
2
1
cos a cos b = &eacute;&euml; cos (a - b ) + cos (a + b ) &ugrave;&ucirc;
2
1
sin a cos b = &euml;&eacute;sin (a + b ) + sin (a - b ) &ucirc;&ugrave;
2
1
cos a sin b = &eacute;&euml;sin (a + b ) - sin (a - b ) &ugrave;&ucirc;
2
Sum to Product Formulas
&aelig;a + b &ouml;
&aelig;a - b &ouml;
sin a + sin b = 2 sin &ccedil;
&divide; cos &ccedil;
&divide;
&egrave; 2 &oslash;
&egrave; 2 &oslash;
&aelig;a + b &ouml; &aelig;a - b &ouml;
sin a - sin b = 2 cos &ccedil;
&divide; sin &ccedil;
&divide;
&egrave; 2 &oslash; &egrave; 2 &oslash;
tan (a &plusmn; b ) =
&aelig;a + b &ouml;
&aelig;a - b &ouml;
cos a + cos b = 2 cos &ccedil;
&divide; cos &ccedil;
&divide;
&egrave; 2 &oslash;
&egrave; 2 &oslash;
&aelig;a + b &ouml; &aelig;a - b &ouml;
cos a - cos b = -2sin &ccedil;
&divide; sin &ccedil;
&divide;
&egrave; 2 &oslash; &egrave; 2 &oslash;
Cofunction Formulas
&aelig;p
&ouml;
sin &ccedil; - q &divide; = cos q
&egrave;2
&oslash;
&aelig;p
&ouml;
csc &ccedil; - q &divide; = sec q
&egrave;2
&oslash;
&aelig;p
&ouml;
cos &ccedil; - q &divide; = sin q
&egrave;2
&oslash;
&aelig;p
&ouml;
sec &ccedil; - q &divide; = csc q
&egrave;2
&oslash;
&aelig;p
&ouml;
tan &ccedil; - q &divide; = cot q
&egrave;2
&oslash;
&aelig;p
&ouml;
cot &ccedil; - q &divide; = tan q
&egrave;2
&oslash;
&copy; 2005 Paul Dawkins
Unit Circle
y
&aelig;
3 1&ouml;
&ccedil;- , &divide;
&egrave; 2 2&oslash;
3p
4
( 0,1)
p
2
&aelig; 1 3&ouml;
&ccedil;- , &divide;
&egrave; 2 2 &oslash;
&aelig;
2 2&ouml;
,
&ccedil;&divide;
&egrave; 2 2 &oslash;
Inverse Trig Functions
2p
3
p
3
90&deg;
120&deg;
45&deg;
30&deg;
p 180&deg;
y = cos -1 x is equivalent to x = cos y
sin ( sin -1 ( x ) ) = x
sin - 1 (sin (q ) ) = q
tan ( tan - 1 ( x ) ) = x
tan - 1 ( tan (q ) ) = q
y = tan x is equivalent to x = tan y
&aelig; 2 2&ouml;
&ccedil;&ccedil; 2 , 2 &divide;&divide;
&egrave;
&oslash;
p
6
Domain and Range
Function
Domain
&aelig; 3 1&ouml;
&ccedil;&ccedil; , &divide;&divide;
&egrave; 2 2&oslash;
y = sin -1 x
-1 &pound; x &pound; 1
y = cos x
-1 &pound; x &pound; 1
y = tan -1 x
-&yen; &lt; x &lt; &yen;
-1
150&deg;
( -1,0)
Inverse Properties
cos ( cos -1 ( x ) ) = x
cos - 1 ( cos (q ) ) = q
-1
p
4
60&deg;
135&deg;
5p
6
&aelig;1 3&ouml;
&ccedil;&ccedil; 2 , 2 &divide;&divide;
&egrave;
&oslash;
Definition
y = sin -1 x is equivalent to x = sin y
0&deg;
0
360&deg;
2p
(1,0)
&aelig;
3 1&ouml;
&ccedil; - ,- &divide;
2&oslash;
&egrave; 2
7p
6
&aelig;
2
2&ouml;
,&ccedil;&divide;
2 &oslash;
&egrave; 2
330&deg;
225&deg;
5p
4
4p
3
240&deg;
&aelig; 1
3&ouml;
&ccedil; - ,&divide;
&egrave; 2 2 &oslash;
270&deg;
3p
2
315&deg;
7p
300&deg;
4
5p
3
( 0,-1)
11p
6
&aelig;1
3&ouml;
&ccedil; ,&divide;
&egrave;2 2 &oslash;
For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y
a
g
Law of Sines
sin a sin b sin g
=
=
a
b
c
Law of Tangents
a - b tan 12 (a - b )
=
a + b tan 12 (a + b )
Law of Cosines
a 2 = b 2 + c 2 - 2bc cos a
b - c tan 12 ( b - g )
=
b + c tan 12 ( b + g )
c = a + b - 2ab cos g
2
&aelig; 5p &ouml; 1
cos &ccedil;
&divide;=
&egrave; 3 &oslash; 2
b
b
b 2 = a 2 + c 2 - 2ac cos b
Example
tan -1 x = arctan x
a
&aelig; 3 1&ouml;
&ccedil; ,- &divide;
&egrave; 2 2&oslash;
&aelig; 2
2&ouml;
,&ccedil;
&divide;
2 &oslash;
&egrave; 2
cos -1 x = arccos x
Law of Sines, Cosines and Tangents
x
c
210&deg;
Alternate Notation
sin -1 x = arcsin x
Range
p
p
- &pound; y&pound;
2
2
0&pound; y&pound;p
p
p
- &lt; y&lt;
2
2
2
2
a - c tan 12 (a - g )
=
a + c tan 12 (a + g )
Mollweide’s Formula
a + b cos 12 (a - b )
=
c
sin 12 g
3
&aelig; 5p &ouml;
sin &ccedil;
&divide;=3
2
&egrave;
&oslash;
&copy; 2005 Paul Dawkins
&copy; 2005 Paul Dawkins
```