x x θ = = 1 sec x θ = tan y x θ = cot x y θ = Facts and Properties. Domain. The
domain is all the values of ... θ x y. 1. © 2005 Paul Dawkins. Formulas and
Identities.
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
y
opposite
1
q x
x
q
tan 2 q + 1 = sec 2 q
adjacent opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent
sin q =
hypotenuse opposite hypotenuse sec q = adjacent adjacent 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 1ö æ q ¹ ç n + ÷ p , n = 0, ± 1, ± 2, K 2ø è q ¹ n p , n = 0, ± 1, ± 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 ) ®
T=
1ö æ q ¹ ç n + ÷ p , n = 0, ± 1, ± 2, K 2ø è q ¹ n p , n = 0, ± 1, ± 2, K
cos (wq ) ®
T
tan (wq ) ®
T
Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 sec q ³ 1 and sec q £ -1 -1 £ cos q £ 1 -¥ < tan q < ¥ -¥ < cot q < ¥
csc (wq ) ®
T
sec ( wq ) ®
T
cot ( wq ) ®
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
© 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 Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then p t px 180t = Þ t= and x = 180 x 180 p
Half Angle Formulas (alternate form) q 1 - cos q 1 sin = ± sin 2 q = (1 - cos ( 2q ) ) 2 2 2 cos
q 1 + cos q =± 2 2
cos 2 q =
1 (1 + cos ( 2q ) ) 2 1 - cos ( 2q )
q 1 - cos q =± tan 2 q = 2 1 + cos q 1 + cos ( 2q ) Sum and Difference Formulas sin (a ± b ) = sin a cos b ± cos a sin b tan
cos (a ± b ) = cos a cos b m sin a sin b tan a ± tan b 1 m tan a tan b Product to Sum Formulas 1 sin a sin b = éëcos (a - b ) - cos (a + b ) ùû 2 1 cos a cos b = éë cos (a - b ) + cos (a + b ) ùû 2 1 sin a cos b = ëésin (a + b ) + sin (a - b ) ûù 2 1 cos a sin b = éësin (a + b ) - sin (a - b ) ùû 2 Sum to Product Formulas æa + b ö æa - b ö sin a + sin b = 2 sin ç ÷ cos ç ÷ è 2 ø è 2 ø æa + b ö æa - b ö sin a - sin b = 2 cos ç ÷ sin ç ÷ è 2 ø è 2 ø tan (a ± b ) =
æa + b ö æa - b ö cos a + cos b = 2 cos ç ÷ cos ç ÷ è 2 ø è 2 ø æa + b ö æa - b ö cos a - cos b = -2sin ç ÷ sin ç ÷ è 2 ø è 2 ø Cofunction Formulas æp ö sin ç - q ÷ = cos q è2 ø æp ö csc ç - q ÷ = sec q è2 ø
æp ö cos ç - q ÷ = sin q è2 ø æp ö sec ç - q ÷ = csc q è2 ø
æp ö tan ç - q ÷ = cot q è2 ø
æp ö cot ç - q ÷ = tan q è2 ø
© 2005 Paul Dawkins
Unit Circle y
æ 3 1ö ç- , ÷ è 2 2ø
3p 4
( 0,1)
p 2
æ 1 3ö ç- , ÷ è 2 2 ø æ 2 2ö , ç÷ è 2 2 ø
Inverse Trig Functions
2p 3
p 3
90° 120°
45° 30°
p 180°
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
æ 2 2ö çç 2 , 2 ÷÷ è ø
p 6
Domain and Range Function Domain
æ 3 1ö çç , ÷÷ è 2 2ø
y = sin -1 x
-1 £ x £ 1
y = cos x
-1 £ x £ 1
y = tan -1 x
-¥ < x < ¥
-1
150° ( -1,0)
Inverse Properties cos ( cos -1 ( x ) ) = x cos - 1 ( cos (q ) ) = q
-1
p 4
60°
135°
5p 6
æ1 3ö çç 2 , 2 ÷÷ è ø
Definition y = sin -1 x is equivalent to x = sin y
0°
0
360°
2p
(1,0)
æ 3 1ö ç - ,- ÷ 2ø è 2
7p 6
æ 2 2ö ,ç÷ 2 ø è 2
330° 225°
5p 4
4p 3
240°
æ 1 3ö ç - ,÷ è 2 2 ø
270°
3p 2
315° 7p 300° 4 5p 3
( 0,-1)
11p 6
æ1 3ö ç ,÷ è2 2 ø
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
æ 5p ö 1 cos ç ÷= è 3 ø 2
b
b
b 2 = a 2 + c 2 - 2ac cos b
Example
tan -1 x = arctan x
a
æ 3 1ö ç ,- ÷ è 2 2ø
æ 2 2ö ,ç ÷ 2 ø è 2
cos -1 x = arccos x
Law of Sines, Cosines and Tangents
x
c 210°
Alternate Notation sin -1 x = arcsin x
Range p p - £ y£ 2 2 0£ y£p p p - < y< 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 æ 5p ö sin ç ÷=3 2 è ø
© 2005 Paul Dawkins
© 2005 Paul Dawkins