Chapter 5 Formulas and Definitions:
(from 5.1)
Fundamental Trigonometric Identities:
Reciprocal Identities
1
cscu
1
cscu =
sin u
Quotient Identities
sin u =
1
secu
1
secu =
cosu
cosu =
1
cot u
1
cotu =
tan u
tan u =
sin u
cosu
cotu =
cosu
sin u
Pythagorean Identities
tan u =
sin 2 u + cos 2 u = 1
Cofunction Identities
1 + tan 2 u = sec 2 u
#!
&
sin % " u ( = cosu
$2
'
#!
&
cos % " u ( = sin u
$2
'
#!
&
tan % " u ( = cot u
$2
'
#!
&
cot % " u ( = tan u
$2
'
1 + cot 2 u = csc 2 u
(from 5.2)
Guidelines for Verifying Trigonometric Identities:
1. Work with one side of the equation at a time. It is often better to
work with the more complicated side first.
2. Look for opportunities to factor an expression, add fractions, square a
binomial, or create a monomial denominator.
3. Look for opportunities to use the fundamental identities. Note which
functions are in the final expression you want. Sines and cosines pair
up well, as do secants and tangents, and cosecants and cotangents.
4. If the preceding guidelines do not help, try converting all terms to sines
and cosines.
5. Always try something. Even making an attempt that leads to a dead
end provides insight.
(from 5.4)
Sum and Difference Formulas:
sin(u + v) = sin u cos v + cosu sin v
sin(u ! v) = sin u cos v ! cosu sin v
cos(u + v) = cosu cos v ! sin u sin v
cos(u ! v) = cosu cos v + sin u sin v
tan u + tan v
1 - tan u tan v
tan u ! tan v
tan(u + v) =
1 + tan u tan v
tan(u + v) =
(from 5.5)
Double-Angle Formulas:
sin(2u) = 2 sin u cosu
cos(2u) = cos 2 u ! sin 2 u
= 2 cos 2 u ! 1
tan(2u) =
2 tan u
1 - tan 2 u
= 1 - 2 sin 2 u
Power-Reducing Formulas:
1 ! cos 2u
1 + cos 2u
sin 2 u =
cos 2u =
2
2
Half-Angle Formulas:
u
1 ! cosu
sin = ±
2
2
tan
cos
tan 2u =
1 ! cos 2u
1 + cos 2u
u
1 + cosu
=±
2
2
u 1 ! cosu
sin u
=
=
2
sin u
1 + cosu
The signs of sin
u
u
u
and cos depend on the quadrant in which lies.
2
2
2
Product-to-Sum Formulas:
1
sin u sin v = ( cos(u ! v) ! cos(u + v))
2
1
cosu cos v = ( cos(u ! v) + cos(u + v))
2
1
sin u cos v = ( sin(u + v) + sin(u ! v))
2
1
cosu sin v = ( sin(u + v) ! sin(u ! v))
2
Sum-to-Product Formulas:
! u + v$
! u ' v$
sin u + sin v = 2 sin #
cos #
&
" 2 %
" 2 &%
! u + v$ ! u ' v$
sin u ' sin v = 2 cos #
sin
" 2 &% #" 2 &%
! u + v$
! u ' v$
cosu + cos v = 2 cos #
cos #
" 2 &%
" 2 &%
! u + v$ ! u ' v$
cosu ' cos v = '2 sin #
sin
" 2 &% #" 2 &%