TRIGONOMETRY
Definition of the Six Trigonometric Functions
Opposite
Right triangle definitions, where 0 < < 2.
opp
hyp
sin csc e
s
u
hyp
opp
en
pot
Hy
adj
hyp
cos sec θ
hyp
adj
Adjacent
opp
adj
tan cot adj
opp
Circular function definitions, where is any angle.
y
r
y
sin csc r = x2 + y2
r
y
(x, y)
x
r
r
cos sec θ
y
r
x
x
y
x
x
cot tan x
y
Reciprocal Identities
1
sin x csc x
1
csc x sin x
1
sec x cos x
1
cos x sec x
sin x
cos x
cot x cos x
sin x
Pythagorean Identities
sin2 x cos2 x 1
1 tan2 x sec2 x
(− 12 , 23 ) π (0, 1) ( 12 , 23 )
90°
(− 22 , 22 ) 3π 23π 2 π3 π ( 22 , 22 )
120°
60°
4 π
45°
(− 23 , 12) 56π 4150°135°
( 23 , 21)
6
30°
(− 1, 0) π 180°
210°
330°
1 cot2 x csc2 x
sin 2u 2 sin u cos u
cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
2 tan u
tan 2u 1 tan2 u
Power-Reducing Formulas
1 cos 2u
2
1 cos 2u
2
cos u 2
1
cos
2u
tan2 u 1 cos 2u
sin2 u Sum-to-Product Formulas
2 x cos x
csc x sec x
2
sec x csc x
2
sin u sin v 2 sin
2 x sin x
tan x cot x
2
cot x tan x
2
cos
Reduction Formulas
sinx sin x
cscx csc x
secx sec x
x
(− 23 , − 12) 76π 5π 225°240° 300°315°7π 116π ( 23 , − 21)
(− 22 , − 22 ) 4 43π 270° 32π 53π 4 ( 22 , − 22 )
1
3
(0, − 1) ( 2 , − 2 )
(− 12 , − 23 )
Cofunction Identities
sin
0° 0
360° 2π (1, 0)
Double -Angle Formulas
1
tan x cot x
1
cot x tan x
Tangent and Cotangent Identities
tan x y
cosx cos x
tanx tan x
cotx cot x
Sum and Difference Formulas
sinu ± v sin u cos v ± cos u sin v
cosu ± v cos u cos v sin u sin v
tan u ± tan v
tanu ± v 1 tan u tan v
© Houghton Mifflin Company, Inc.
u 2 v cosu 2 v
uv
uv
sin u sin v 2 cos
sin
2 2 uv
uv
cos u cos v 2 cos
cos
2 2 uv
uv
cos u cos v 2 sin
sin
2 2 Product-to-Sum Formulas
1
sin u sin v cosu v cosu v
2
1
cos u cos v cosu v cosu v
2
1
sin u cos v sinu v sinu v
2
1
cos u sin v sinu v sinu v
2