TRIGONOMETRY

Right Triangle Definitions Circular Definitions Other Identities

sin cos

tan

sec csc

opp adj

hyp hypopp adj

cotadj opphyp hyp

adj opp

θ θ

θ θ

θ θ

= =

= =

= =

sin cos

tan

sec csc

y x

r ry x

cotx yr r

x y

θ θ

θ θ

θ θ

= =

= =

= =

sin costan cot

cos sin1 1

sec csccos sin

x xx x

x x

x xx x

= =

= =

Reduction Formulas Sum and Difference Formulas Pythagorean Identities

sin( ) sin( ) cos( ) cos( )tan( ) tan( ) cot( ) cot( )sec( ) sec( ) csc( ) csc( )

x x x xx x x xx x x

− = − − =− = − − = −− = − = − x

cos( ) cos cos sin sinsin( ) sin cos cos sin

tan tantan( )

1 tan tan

u v u v u vu v u v u v

u vu v

u v

± = ⋅ ⋅± = ⋅ ± ⋅

±± =

⋅

∓

∓

2 2

2 2

2 2

sin cos 1

1 tan sec

1 cot csc

θ θ

θ θ

θ θ

+ =

+ =

+ =

Double Angle Formulas Power Reducing Formulas Cofunction Identities

2 2

2

2

2

sin(2 ) 2 sin cos

cos(2 ) cos sin

cos(2 ) 2 cos 1

cos(2 ) 1 2 sin2 tan

tan(2 )1 tan

u u u

u u

u u

u uu

uu

=

= −

= −

= −

=−

u

2

2

2

1 cos(2 )sin

21 cos(2 )

cos2

1 cos(2 )tan

1 cos(2 )

uu

uu

uu

u

−=

+=

−=

+

sin cos cos sin2 2

tan cot cot tan2 2

sec csc csc sec2 2

x x x x

x x x x

x x x

π π

π π

π π

− = − =

− = − =

− = − =

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x

Product to Sum Formulas Special Angles sin sin 0.5[cos( ) cos( )]cos cos 0.5[cos( ) cos( )]sin cos 0.5[sin( ) sin( )]cos sin 0.5[sin( ) sin( )]

u v u v u vu v u v u vu v u v u vu v u v u v

= − − += − + += + + −= + − −

3 2 1cos 0 1 cos cos cos cos 0

6 2 4 2 3 2 21 2 3

sin 0 0 sin sin sin sin 16 2 4 2 3 2 2

π π π

π π π π

π= = = = =

= = = = =

Derivative Rules

2 2

sin cos cos sin

tan sec cot csc

sec sec tan csc csc cot

d dx x x x

dx dxd d

x x x xdx dxd d

x x x x xdx dx

= = −

= = −

= = − x

2

2

2

1arcsin

11

arctan1

1arc sec

1

dx

dx xd

xdx xd

xdx x x

=−

=+

=−

1ln

1cosh

sinh

ln

sinh

cosh

x xax

n nx

d dx a a

dx dxd d

x x n xdx dxd

x xdx

−

= = ⋅

= =

=

⋅