1230 words - 5 pages

© 2005 Paul Dawkins

Trig Cheat Sheet

Definition of the Trig Functions Right triangle definition For this definition we assume that

0 2 p

q< < or 0 90q° < < ° .

oppositesin

hypotenuse q = hypotenusecsc

opposite q =

adjacentcos hypotenuse

q = hypotenusesec adjacent

q =

oppositetan adjacent

q = adjacentcot opposite

q =

Unit circle definition For this definition q is any angle.

sin

1 y yq = = 1csc

y q =

cos 1 x xq = = 1sec

x q =

tan y x

q = cot x y

q =

Facts and Properties Domain The domain is all the values of q that can be plugged into the function. sinq , q can be any angle cosq , q can be any angle

tanq , 1 , 0, 1, 2, 2

n nq pæ ö¹ + = ± ±ç ÷ è ø

K

cscq , , 0, 1, 2,n nq p¹ = ± ± K

secq , 1 , 0, 1, 2, 2

n nq pæ ö¹ + = ± ±ç ÷ è ø

K

cotq , , 0, 1, 2,n nq p¹ = ± ± K Range The range is all possible values to get out of the function.

1 sin 1q- £ £ csc 1 and csc 1q q³ £ - 1 cos 1q- £ £ sec 1 andsec 1q q³ £ -

tanq-¥ < < ¥ cotq-¥ < < ¥

Period The period of a function is the number, T, such that ( ) ( )f T fq q+ = . So, if w is a fixed number and q is any angle we have the following periods.

( )sin wq ® 2T p w

=

( )cos wq ® 2T p w

=

( )tan wq ® T p w

=

( )csc wq ® 2T p w

=

( )sec wq ® 2T p w

=

( )cot wq ® T p w

=

q adjacent

opposite hypotenuse

x

y

( ),x y

q

x

y 1

© 2005 Paul Dawkins

Formulas and Identities Tangent and Cotangent Identities

sin costan cot cos sin

q q q q

q q = =

Reciprocal Identities 1 1csc sin

sin csc 1 1sec cos

cos sec 1 1cot tan

tan cot

q q q q

q q q q

q q q q

= =

= =

= =

Pythagorean Identities 2 2

2 2

2 2

sin cos 1 tan 1 sec 1 cot csc

q q

q q

q q

+ =

+ =

+ =

Even/Odd Formulas ( ) ( ) ( ) ( ) ( ) ( )

sin sin csc csc

cos cos sec sec

tan tan cot cot

q q q q

q q q q

q q q q

- = - - = -

- = - =

- = - - = -

Periodic Formulas If n is an integer.

( ) ( ) ( ) ( ) ( ) ( )

sin 2 sin csc 2 csc

cos 2 cos sec 2 sec

tan tan cot cot

n n

n n

n n

q p q q p q

q p q q p q

q p q q p q

+ = + =

+ = + =

+ = + = Double Angle Formulas

( ) ( )

( )

2 2

2

2

2

sin 2 2sin cos

cos 2 cos sin

2cos 1 1 2sin

2 tantan 2 1 tan

q q q

q q q

q

q q

q q

=

= -

= -

= -

= -

Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then

180and 180 180

t x tt x x

p p p

= Þ = =

Half Angle Formulas

( )( )

( )( ) ( ) ( )

2

2

2

1sin 1 cos 2 2 1cos 1 cos 2 2 1 cos 2

tan 1 cos 2

q q

q q

q q

q

= -

= +

- =

+

Sum and Difference Formulas ( ) ( )

( )

sin sin cos cos sin

cos cos cos sin sin tan tantan

1 tan tan

a b a b a b

a b a b a b

a b a b

a b

± = ±

± =

± ± =

m

m

Product to Sum Formulas

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1sin sin cos cos 2 1cos cos cos cos 2 1sin cos sin sin 2 1cos sin sin sin 2

a b a b a b

a b a b a b

a b a b a b

a b a b a b

= - -...

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