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 < 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


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

