olga lednichenko, Trigonometry, Calculus, Algebra,

1. Trig Cheat Sheet Formulas and Identities
Tangent and Cotangent Identities Half Angle Formulas
sin q cos q 1
Definition of the Trig Functions tan q = cot q = sin 2 q = (1 - cos ( 2q ) )
Right triangle definition cos q sin q 2
For this definition we assume that Reciprocal Identities 1
Unit circle definition
1 1 cos 2 q = (1 + cos ( 2q ) )
p For this definition q is any angle. csc q = sin q = 2
0 < q < or 0° < q < 90° .
2 y sin q csc q 1 - cos ( 2q )
1 1 tan q =
2
sec q = cos q = 1 + cos ( 2q )
( x, y ) cos q sec q
Sum and Difference Formulas
1 1 1
hypotenuse y q cot q = tan q = sin (a ± b ) = sin a cos b ± cos a sin b
opposite tan q cot q
x
x
Pythagorean Identities cos (a ± b ) = cos a cos b m sin a sin b
q sin 2 q + cos 2 q = 1 tan a ± tan b
tan (a ± b ) =
adjacent tan q + 1 = sec q
2 2 1 m tan a tan b
Product to Sum Formulas
opposite hypotenuse y 1 1 + cot 2 q = csc 2 q
sin q = csc q = sin q = =y csc q = 1
sin a sin b = é cos (a - b ) - cos (a + b ) ù
hypotenuse opposite 1 y Even/Odd Formulas 2ë û
adjacent hypotenuse x 1 sin ( -q ) = - sin q csc ( -q ) = - csc q 1
cos q = secq = cos q = = x sec q = cos a cos b = é cos (a - b ) + cos (a + b ) ù
hypotenuse adjacent 1 x cos ( -q ) = cos q sec ( -q ) = sec q 2ë û
opposite adjacent y x 1
tan q = cot q = tan q = cot q = tan ( -q ) = - tan q cot ( -q ) = - cot q sin a cos b = ésin (a + b ) + sin (a - b ) ù
adjacent opposite x y 2ë û
Periodic Formulas 1
Facts and Properties cos a sin b = ésin (a + b ) - sin (a - b ) ù
2ë û
If n is an integer.
Domain sin (q + 2p n ) = sin q csc (q + 2p n ) = csc q Sum to Product Formulas
The domain is all the values of q that Period
cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q æa + b ö æa - b ö
can be plugged into the function. The period of a function is the number, sin a + sin b = 2 sin ç ÷ cos ç ÷
T, such that f (q + T ) = f (q ) . So, if w tan (q + p n ) = tan q cot (q + p n ) = cot q è 2 ø è 2 ø
sin q , q can be any angle
is a fixed number and q is any angle we Double Angle Formulas æa + b ö æa - b ö
sin a - sin b = 2 cos ç ÷ sin ç ÷
cos q , q can be any angle have the following periods. è 2 ø è 2 ø
æ 1ö sin ( 2q ) = 2sin q cos q
tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K æa + b ö æa - b ö
2p cos ( 2q ) = cos 2 q - sin 2 q cos a + cos b = 2 cos ç ÷ cos ç ÷
è 2ø
sin ( wq ) ® T = è 2 ø è 2 ø
csc q , q ¹ n p , n = 0, ± 1, ± 2,K w = 2 cos 2 q - 1 æa + b ö æa - b ö
æ 1ö 2p cos a - cos b = -2 sin ç ÷ sin ç ÷
sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K cos (wq ) ® T = = 1 - 2sin 2 q è 2 ø è 2 ø
è 2ø w
2 tan q Cofunction Formulas
cot q , q ¹ n p , n = 0, ± 1, ± 2,K tan (wq ) ® T =
p tan ( 2q ) =
1 - tan 2 q æp ö æp ö
w sin ç - q ÷ = cos q cos ç - q ÷ = sin q
Range 2p Degrees to Radians Formulas è2 ø è2 ø
csc (wq ) ® T =
The range is all possible values to get w If x is an angle in degrees and t is an æp ö æp ö
csc ç - q ÷ = sec q sec ç - q ÷ = csc q
out of the function. 2p angle in radians then è2 ø è2 ø
-1 £ sin q £ 1 csc q ³ 1 and csc q £ -1 sec (wq ) ® T = p t px 180t
w = Þ t= and x = æp ö æp ö
tan ç - q ÷ = cot q cot ç - q ÷ = tan q
-1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 p 180 x 180 p è 2 ø è 2 ø
-¥ < tan q < ¥ -¥ < cot q < ¥ cot (wq ) ® T =
w
© 2005 Paul Dawkins © 2005 Paul Dawkins

2. Unit Circle
Inverse Trig Functions
Definition Inverse Properties
y
( 0,1) y = sin - 1 x is equivalent to x = sin y cos ( cos -1 ( x ) ) = x cos -1 ( cos (q ) ) = q
p y = cos - 1 x is equivalent to x = cos y sin ( sin -1 ( x ) ) = x sin -1 ( sin (q ) ) = q
æ1 3ö
æ 1 3ö ç , ÷
ç2 2 ÷
ç- , ÷ 2 è ø
y = tan x is equivalent to x = tan y
-1
è 2 2 ø p æ 2 2ö tan ( tan - 1 ( x ) ) = x tan -1 ( tan (q ) ) = q
2p 90° ç 2 , 2 ÷
ç ÷
æ 2 2ö 3 è ø
ç- , ÷ 3 Domain and Range
è 2 2 ø 120° 60°
p Alternate Notation
3p æ 3 1ö Function Domain Range
4 ç 2 ,2 ÷
ç sin -1 x = arcsin x
4 è
÷
ø p p
æ 3 1ö
ç- , ÷ 135° 45° p y = sin -1 x -1 £ x £ 1 - £ y£
è 2 2ø 5p 2 2 cos - 1 x = arccos x
6 y = cos x -1
-1 £ x £ 1 0£ y £p
6 30° tan - 1 x = arctan x
150° p p
y = tan -1 x -¥ < x < ¥ - < y<
2 2
( -1,0 ) p 180° 0° 0 (1,0 )
x Law of Sines, Cosines and Tangents
360° 2p
c b a
210°
7p 330°
11p
6 225°
æ 3 1ö 6 æ 3 1ö a g
ç - ,- ÷ 5p 315° ç ,- ÷
è 2 2ø è 2 2ø
4 240° 300° 7p
æ 2 2ö 4p 270°
4
b
ç- ,- ÷ 5p æ 2 2ö
è 2 2 ø 3 3p ç ,- ÷
3 è 2 2 ø
2 Law of Sines Law of Tangents
æ 1 3ö
ç - ,- ÷
æ1
ç ,-
3ö
÷ sin a sin b sin g
= = a - b tan 1 (a - b )
è 2 2 ø è2 2 ø = 2
( 0,-1) a b c a + b tan 1 (a + b )
2
Law of Cosines b - c tan 1 ( b - g )
= 2
a 2 = b 2 + c 2 - 2bc cos a b + c tan 1 ( b + g )
2
For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y b 2 = a 2 + c 2 - 2ac cos b a - c tan 1 (a - g )
= 2
c = a + b - 2ab cos g a + c tan 1 (a + g )
2 2 2
2
Example
Mollweide’s Formula
æ 5p ö 1 æ 5p ö 3
cos ç ÷= sin ç ÷=- a + b cos 1 (a - b )
è 3 ø 2 è 3 ø 2 = 2
c sin 1 g
2
© 2005 Paul Dawkins © 2005 Paul Dawkins