Trig Sheet 1 Essay

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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