Trigonometry Formulas

TRIGONOMETRY
Right Triangle Definitions
opp
adj
sin θ =
cos θ =
hyp
hyp
opp
adj
cotθ =
tan θ =
adj
opp
hyp
hyp
sec θ =
csc θ =
adj
opp
Circular Definitions
sin θ =
tan θ =
sec θ =
y
cos θ =
r
y
cotθ =
x
r
csc θ =
x
Other Identities
x
r
x
tan x =
y
r
sec x =
sin x
cot x =
cos x
1
cos x
sin x
1
csc x =
cos x
sin x
y
Reduction Formulas
Sum and Difference Formulas
Pythagorean Identities
sin( − x ) = − sin( x )
cos(u ± v ) = cos u ⋅ cos v ∓ sin u ⋅ sin v
sin(u ± v ) = sin u ⋅ cos v ± cos u ⋅ sin v
tan u ± tan v
tan(u ± v ) =
1 ∓ tan u ⋅ tan v
sin θ + cos θ = 1
cos( − x ) = cos( x )
tan( − x ) = − tan( x ) cot( − x ) = − cot( x )
sec( − x ) = sec( x )
csc( − x ) = − csc( x )
Double Angle Formulas
sin(2u ) = 2 sin u cos u
2
2
cos(2u ) = cos u − sin u
Power Reducing Formulas
2
1 − cos(2u )
2
2
1 + cos(2u )
2
2
1 − cos(2u )
sin u =
2
cos(2u ) = 2 cos u − 1
cos u =
2
cos(2u ) = 1 − 2 sin u
tan(2u ) =
2 tan u
tan u =
2
1 − tan u
cos u cos v = 0.5[cos(u − v ) + cos(u + v )]
sin u cos v = 0.5[sin(u + v ) + sin(u − v )]
2
2
2
2
1 + tan θ = sec θ
1 + cot θ = csc θ
⎛π
⎞
⎛π
⎞
− x ⎟ = cos x cos ⎜ − x ⎟ = sin x
⎝2 ⎠
⎝2 ⎠
⎛π
⎞
⎛π
⎞
tan ⎜ − x ⎟ = cot x cot ⎜ − x ⎟ = tan x
⎝2 ⎠
⎝2 ⎠
⎛π
⎞
⎛π
⎞
sec ⎜ − x ⎟ = csc x csc ⎜ − x ⎟ = sec x
⎝2 ⎠
⎝2 ⎠
sin ⎜
Special Angles
cos 0 = 1 cos
π
3
=
6
sin 0 = 0 sin
cos u sin v = 0.5[sin(u + v ) − sin(u − v )]
2
Cofunction Identities
1 + cos(2u )
Product to Sum Formulas
sin u sin v = 0.5[cos(u − v ) − cos(u + v )]
2
π
cos
2
=
6
1
2
π
=
4
sin
π
=
4
2
2
2
cos
π
=
3
sin
2
π
1
cos
2
=
3
π
=0
2
3
sin
2
π
=1
2
Derivative Rules
d
dx
d
dx
d
dx
sin x = cos x
2
tan x = sec x
sec x = sec x tan x
d
dx
d
dx
d
dx
cos x = − sin x
2
cot x = − csc x
csc x = − csc x cot x
d
dx
d
dx
d
dx
arcsin x =
arctan x =
arc sec x =
1
d
1− x
1
dx
d
2
2
x +1
1
2
x x −1
dx
d
dx
ln x =
1
x
cosh x = sinh x
sinh x = cosh x
d
x
x
a = ln a ⋅ a
dx
d n
n −1
x = n⋅x
dx