Formulas and Identities
Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90 ° . 2
Unit circle definition For this definition q is any angle. y
( x, y ) hypotenuse
1
y
opposite
q x
x
sin 2 q + cos 2 q = 1
q
tan 2 q + 1 = sec 2 q
adjacent
= sin q = = cosq = = tan q =
opposite hypotenuse adjacent hypotenuse opposite adjacent
= csc q = = sec q = = cot q =
hypotenuse opposite hypotenuse adjacent adjacent opposite
= sin q = = cosq = = tan q =
y 1 x 1 y
= y
= cscq =
= x
= sec q =
x
= cot q =
1 y 1 x x 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 , csc q , secq 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 se c q = cos q = cos q sec q 1 1 cot q = tan q = tan q cot q Pythagorean Identities
q can be any angle q can be any angle 1ö æ q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è q ¹ n p , n = 0, ± 1, ± 2,K
1ö æ q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è q ¹ n p , n = 0, ± 1, ± 2,K
Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ s in q £ 1 sec q ³ 1 and sec q £ -1 -1 £ cos q £ 1 -¥ < tan q < ¥ -¥ < cot q < ¥
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. sin ( wq ) ®
T =
cos (wq ) ®
T =
tan (wq ) ®
T =
csc ( wq ) ®
T =
sec ( wq ) ®
T =
cot ( wq ) ®
T =
2p w 2p w p w 2p w 2p w p w
© 2005 Paul Dawkins
Half Angle Formulas
q
sin
2 q
cos tan
2 q 2
=± =± =±
1 - cos q 2 1 + cos q 2 1 - cos q 1 + cos q
(alternate form)
1
2 sin q =
2 1
2 cos q =
tan q =
2
(1 - cos ( 2q ) ) (1 + cos ( 2q ) )
1 - cos ( 2q )
2
1 + cos ( 2q )
Sum and Difference Formulas sin ( a ± b ) = sin a cos b ± cos a sin b
cos (a ± b ) = cos a cos b tan (a ± b ) =
m sin a
sin b
tan a ± tan b
1 m tan a tan b Product to Sum Formulas 1 Even/Odd Formulas sin a sin b = éëcos (a - b ) - cos (a + b ) ùû 2 sin ( -q ) = - sin q csc ( -q ) = - csc q 1 cos a cos b = éëcos (a - b ) + cos (a + b ) ùû cos ( -q ) = cos q sec ( -q ) = sec q 2 tan ( -q ) = - tan q cot ( -q ) = - cot q 1 sin a cos b = éësin (a + b ) + sin (a - b ) ùû 2 Periodic Formulas 1 If n is an integer. cos a sin b = éësin (a + b ) - sin (a - b ) ùû 2 sin (q + 2p n ) = sin q csc (q + 2p n ) = csc q Sum to Product Formulas cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q æa + b ö æ a - b ö sin a + sin b = 2 s in ç ÷ cos ç ÷ tan (q + p n ) = tan q cot (q + p n ) = cot q è 2 ø è 2 ø Double Angle Formulas æ a + b ö æ a - b ö sin a - sin b = 2 c os ç ÷ sin ç ÷ sin ( 2q ) = 2 sin q cos q è 2 ø è 2 ø 1 + cot q = csc q 2
2
cos ( 2q ) = cos 2 q - sin 2 q
= 2 cos2 q - 1 = 1 - 2 sin 2 q tan ( 2q ) =
2tan q 1 - tan 2 q
Degrees to Radians Formulas If x If x is is an angle in degrees and t is is an angle in radians then p t p x 180t = Þ t= and x = 180 x 180 p
æa + b ö æ a - b ö ÷ cos ç ÷ è 2 ø è 2 ø æ a + b ö æ a - b ö cos a - cos b = - 2 sin ç ÷ sin ç ÷ è 2 ø è 2 ø cos a + cos b = 2 c os ç
Cofunction Formulas æp ö - q ÷ = cos q è2 ø
cos ç
æp ö - q ÷ = s ec q è2 ø
sec ç
æp ö - q ÷ = cot q è2 ø
cot ç
sin ç
csc ç
tan ç
æ p -q è2
ö ÷ = sin q ø
æ p -q è2
ö ÷ = cscq ø
æ p ö - q ÷ = t an q è2 ø
© 2005 Paul Dawkins
Unit Circle
y
2 2p
æ 2 2ö , ç÷ è 2 2 ø
3
æ 2 2ö çç 2 , 2 ÷÷ è ø
3
x is equivalent to x = sin y
y = cos
-1
x is equivalent to x = cos y
y = tan
-1
x is equivalent to x = tan y
Domain and Range Function Domain
p
60 °
4 45 °
135 °
5p
p 30 °
6
( -1,0)
p
90 °
120 °
3p 4
æ1 3ö çç , ÷÷ è2 2 ø
Inverse Properties
-1
y = sin
( 0,1)
p
æ 1 3ö ç- , ÷ è 2 2 ø
æ 3 1ö ç- , ÷ è 2 2ø
Inverse Trig Functions Definition
æ 3 1ö çç 2 , 2 ÷÷ è ø
y = sin
6
-1
x
- 1 £ x £ 1
y = cos x
- 1 £ x £ 1
-1
150 °
- ¥ < x < ¥
y = tan x -1
0°
0
360 °
2p
p 180 °
(1,0)
æ 3 1ö ç - ,- ÷ è 2 2ø
210 °
6
330 ° 225 °
5p
-
p 2
< y <
æ 2 2ö ,ç÷ 2 ø è 2
4
240 °
4p
270 °
3
3p 2
æ 1 3ö ç - ,÷ è 2 2 ø
7p
300 °
5p
4
3
æ 3 1ö ç ,- ÷ è 2 2ø
tan ( tan
( x ) ) = x
-1
( x ) ) = x
-1
(cos (q ) ) = q sin ( sin (q ) ) = q tan - ( tan (q ) ) = q
cos
-1
1
Alternate Notation -1
sin x = arcsin x -1
cos x = arccos x -
tan 1 x = arctan x
p 2
b
a
æ 2 2ö ,ç ÷ 2 ø è 2
æ1 3ö ç ,÷ è2 2 ø
a
Law of Sines sin a sin b
=
a
=
sin g
b
c
a +b b-c
a = b + c - 2bc cos a
b+c a-c
c = a + b - 2ab cos g
a+c
2
Example
Law of Tangents a - b tan 12 (a - b )
b 2 = a 2 + c 2 - 2ac cos b
2
For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y
g
b
Law of Cosines
æ 5p ö 1 ÷= è 3 ø 2
-1
( x ) ) = x
Law of Sines, Cosi nes and Tangents
x
( 0, -1)
cos ç
-1
11p 6
315 °
sin (sin
Range p p - £ y £ 2 2 0 £ y £ p
c 7p
cos ( cos
2
2
2
2
=
tan 12 (a + b )
=
tan 12 ( b - g )
=
tan 12 (a - g )
tan 12 ( b + g ) tan 12 (a + g )
Mollweide’s Formula a + b cos 12 (a - b )
3 æ 5p ö ÷ =2 è 3 ø
sin ç
c
© 2005 Paul Dawkins
=
sin 12 g
© 2005 Paul Dawkins
