Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that Unit circle definition
p For this definition q is any angle.
0 < q < or 0° < q < 90° .
2 y
( x, y )
hypotenuse 1
y q
opposite x
x
q
adjacent
opposite hypotenuse y 1
sin q = csc q = sin q = =y csc q =
hypotenuse opposite 1 y
adjacent hypotenuse x 1
cos q = sec q = cos q = = x sec q =
hypotenuse adjacent 1 x
opposite adjacent y x
tan q = cot q = tan q = cot q =
adjacent opposite x y
Facts and Properties
Domain
The domain is all the values of q that Period
can be plugged into the function. The period of a function is the number,
T, such that f (q + T ) = f (q ) . So, if w
sin q , q can be any angle is a fixed number and q is any angle we
cos q , q can be any angle have the following periods.
æ 1ö
tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K
è 2ø 2p
sin ( wq ) ® T=
csc q , q ¹ n p , n = 0, ± 1, ± 2,K w
æ 1ö 2p
sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K cos (wq ) ® T =
è 2ø w
cot q , q ¹ n p , n = 0, ± 1, ± 2,K p
tan (wq ) ® T =
w
Range 2p
csc (wq ) ® T =
The range is all possible values to get w
out of the function. 2p
-1 £ sin q £ 1 csc q ³ 1 and csc q £ -1 sec (wq ) ® T =
w
-1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 p
-¥ < tan q < ¥ -¥ < cot q < ¥ cot (wq ) ® T =
w
Formulas and Identities
Tangent and Cotangent Identities Half Angle Formulas (alternate form)
sin q cos q q 1 - cos q 1
tan q = cot q = sin = ± sin 2 q = (1 - cos ( 2q ) )
cos q sin q 2 2 2
Reciprocal Identities
q 1 + cos q 1
csc q =
1
sin q =
1 cos
2
=±
2
cos 2 q =
2
(1 + cos ( 2q ) )
sin q csc q
1 1 q 1 - cos q 1 - cos ( 2q )
sec q = cos q = tan = ± tan 2 q =
cos q sec q 2 1 + cos q 1 + cos ( 2q )
1 1 Sum and Difference Formulas
cot q = tan q =
tan q cot q sin (a ± b ) = sin a cos b ± cos a sin b
Pythagorean Identities cos (a ± b ) = cos a cos b m sin a sin b
sin 2 q + cos 2 q = 1
tan a ± tan b
tan 2 q + 1 = sec 2 q tan (a ± b ) =
1 m tan a tan b
1 + cot 2 q = csc 2 q Product to Sum Formulas
1
Even/Odd Formulas sin a sin b = éëcos (a - b ) - cos (a + b ) ùû
sin ( -q ) = - sin q csc ( -q ) = - csc q 2
1
cos ( -q ) = cos q sec ( -q ) = sec q cos a cos b = éë cos (a - b ) + cos (a + b ) ùû
2
tan ( -q ) = - tan q cot ( -q ) = - cot q 1
sin a cos b = éësin (a + b ) + sin (a - b ) ùû
Periodic Formulas 2
If n is an integer. 1
cos a sin b = éësin (a + b ) - sin (a - b ) ùû
sin (q + 2p n ) = sin q csc (q + 2p n ) = csc q 2
Sum to Product Formulas
cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q
æa + b ö æa - b ö
tan (q + p n ) = tan q cot (q + p n ) = cot q sin a + sin b = 2sin ç ÷ cos ç ÷
è 2 ø è 2 ø
Double Angle Formulas
æa + b ö æa - b ö
sin a - sin b = 2 cos ç ÷ sin ç ÷
sin ( 2q ) = 2sin q cos q è 2 ø è 2 ø
cos ( 2q ) = cos 2 q - sin 2 q cos a + cos b = 2 cos ç
æa + b ö æa - b ö
÷ cos ç ÷
= 2 cos 2 q - 1 è 2 ø è 2 ø
æa + b ö æa - b ö
= 1 - 2sin 2 q cos a - cos b = -2sin ç ÷ sin ç ÷
è 2 ø è 2 ø
2 tan q
tan ( 2q ) = Cofunction Formulas
1 - tan 2 q
æp ö æp ö
Degrees to Radians Formulas sin ç - q ÷ = cos q cos ç - q ÷ = sin q
è2 ø è2 ø
If x is an angle in degrees and t is an
angle in radians then æp ö æp ö
csc ç - q ÷ = sec q sec ç - q ÷ = csc q
p t px 180t è2 ø è2 ø
= Þ t= and x = æp ö æp ö
180 x 180 p tan ç - q ÷ = cot q cot ç - q ÷ = tan q
è2 ø è2 ø
Unit Circle
y
( 0,1)
p æ1 3ö
æ 1 3ö çç 2 , 2 ÷÷
ç- , ÷ 2 è ø
è 2 2 ø p æ 2 2ö
2p 90° çç , ÷÷
æ 2 2ö 3 è 2 2 ø
ç- , ÷ 3
è 2 2 ø 120° p
3p 60° æ 3 1ö
4 çç 2 , 2 ÷÷
æ 3 1ö 4 45° p è ø
ç- , ÷ 135°
è 2 2ø 5p
6
6 30°
150°
( -1,0 ) p 180° 0° 0 (1,0 )
360° 2p x
210°
7p 330°
11p
6 225°
æ 3 1ö 6 æ 3 1ö
ç - ,- ÷ 5p 315° ç ,- ÷
è 2 2ø è 2 2ø
4 240° 300° 7p
æ 2ö 4p 270°
ç-
2
,- ÷ 5p 4 æ 2 2ö
è 2 2 ø 3 3p ç
2
,-
2
÷
3 è ø
æ 1 3ö 2 æ
ç - ,- ÷ 1 3ö
2 2 ç ,- ÷
è ø è2 2 ø
( 0,-1)
For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y
Example
æ 5p ö 1 æ 5p ö 3
cos ç ÷= sin ç ÷=-
è 3 ø 2 è 3 ø 2
Inverse Trig Functions
Definition Inverse Properties
y = sin -1 x is equivalent to x = sin y cos ( cos -1 ( x ) ) = x cos -1 ( cos (q ) ) = q
y = cos -1 x is equivalent to x = cos y sin ( sin -1 ( x ) ) = x sin -1 ( sin (q ) ) = q
y = tan -1 x is equivalent to x = tan y
tan ( tan -1 ( x ) ) = x tan -1 ( tan (q ) ) = q
Domain and Range
Function Domain Range Alternate Notation
p p sin -1 x = arcsin x
y = sin -1 x -1 £ x £ 1 - £ y£
2 2 cos -1 x = arccos x
y = cos -1 x -1 £ x £ 1 0£ y £p tan -1 x = arctan x
p p
y = tan -1 x -¥ < x < ¥ - < y<
2 2
Law of Sines, Cosines and Tangents
c b a
a g
Law of Sines Law of Tangents
sin a sin b sin g
= = a - b tan 12 (a - b )
=
a b c a + b tan 12 (a + b )
Law of Cosines b - c tan 12 ( b - g )
=
a 2 = b2 + c 2 - 2bc cos a b + c tan 12 ( b + g )
b 2 = a 2 + c 2 - 2ac cos b a - c tan 12 (a - g )
=
c = a + b - 2ab cos g a + c tan 12 (a + g )
2 2 2
Mollweide’s Formula
a + b cos 12 (a - b )
=
c sin 12 g
