Trig Cheat Sheet

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 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 =y 1 x cos q = = x 1 y tan q = x

sin q =

1 y 1 sec q = x x cot q = y csc q =

Facts and Properties Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , 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 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec 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. 2p w 2p = w p = w 2p = w 2p = w p = w

sin ( wq ) ®

T=

cos (wq ) ®

T

tan (wq ) ®

T

csc (wq ) ®

T

sec (wq ) ®

T

cot (wq ) ®

T

© 2005 Paul Dawkins

Formulas and Identities 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 sec q = 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 tan 2 q + 1 = sec 2 q 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 - 2sin 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) 1 - cos q 1 q sin = ± sin 2 q = (1 - cos ( 2q ) ) 2 2 2 1 + cos q q =± 2 2

1 (1 + cos ( 2q ) ) 2 1 - cos ( 2q ) 1 - cos q q tan = ± 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 cos

cos 2 q =

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 = éësin (a + b ) + sin (a - b ) ùû 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 = 2sin 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 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 tan (a ± b ) =

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

p 2

æ 1 3ö ç- , ÷ è 2 2 ø æ 2 2ö , ç÷ è 2 2 ø æ 3 1ö ç- , ÷ 2 2ø è

( -1,0 )

3p 4

5p 6

( 0,1)

2p 3

p 3

90° 120°

æ1 3ö çç 2 , 2 ÷÷ è ø

p 4

60° 45°

135°

30°

p 6

æ 3 1ö çç 2 , 2 ÷÷ è ø

150°

p 180°

æ 3 1ö ç - ,- ÷ 2 2ø è

æ 2 2ö , çç ÷÷ è 2 2 ø

7p 6

æ 2 2ö ,ç÷ 2 ø è 2

210°

0°

0

360°

2p

330° 225°

5p 4

4p 3

240°

æ 1 3ö ç - ,÷ 2 2 è ø

315° 7p 300° 270° 4 5p 3p 3 2 æ

11p 6

(1,0 )

x

æ 3 1ö ç ,- ÷ è 2 2ø

æ 2 2ö ,ç ÷ 2 2 è ø

1 3ö ç ,÷ 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 cos ç è 3

ö 1 ÷= ø 2

æ 5p sin ç è 3

3 ö ÷=2 ø


Inverse Trig Functions Definition y = sin -1 x is equivalent to x = sin y

Inverse Properties cos ( cos -1 ( x ) ) = x cos -1 ( cos (q ) ) = q

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 y = sin -1 x

-1 £ x £ 1

y = cos -1 x

-1 £ x £ 1

y = tan -1 x

-¥ < x < ¥

sin ( sin -1 ( x ) ) = x

sin -1 ( sin (q ) ) = q

tan ( tan -1 ( x ) ) = x

tan -1 ( tan (q ) ) = q

Alternate Notation sin -1 x = arcsin x

Range p p - £ y£ 2 2 0£ y £p p p - < y< 2 2

cos -1 x = arccos x tan -1 x = arctan x

Law of Sines, Cosines and Tangents c

b

a

a

g

b 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 = b2 + c 2 - 2bc cos a

b - c tan 12 ( b - g ) = b + c tan 12 ( b + g )

b 2 = a 2 + c 2 - 2ac cos b c = a + b - 2ab cos g 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
