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