Tabla de Identidades Trigonometricas 2012 neoparaiso.com/imprimir

Funciones Trigonometricas

tan x =sin x, sec x =1,csc x =1, cot x =1=cos x

cos xcos xsin xtan xsin x

Funciones Trigonometricas en funcion de las Otras Cinco

sin xcos xtan x

tan x

2

sin x =sin x 1 cosx

1 + tan2 x

1

cos x =p1 sin2 xcos x

1 + tan2 x

tan x =sin x1 cos2 xtan x

p1 sin2 xcos x

12x

csc x =11 + tan

sin xtan x

1 cos2 x

11

sec x =1 + tan2 x

pcos x

2

1 sin x

2

cot x =1 sin xcos x1

p sin x

1cos2 xtan x

csc xsec xcot x

sin x =1sec2 x 11

csc xsec x1 + cot2 x

cos x =csc2 x 11cot x

csc xsec x1 + cot2 x

11

2

tan x =secx 1cot x

csc2 x 1

sec x

2

csc x =csc x1 + cotx

sec2 x 1

sec x =csc xsec x1 + cot2 x

cot x

csc2 x 1

1

2

cot x =cscx 1cot x

sec2 x 1

Algunos Valores Especiales

Funcion0(0)(15) (30) (45) (60)5(75) (90)

1212

6432

sin06 21236+21

2224

4

cos16+23216 20

4222

4

@()

tan02 3

3132 +3

3

@()2

csc231

6 +2236 2

2@()

sec132

6 2326 +2

@()

32

cot2 +33130

3

Identidades por Simetra, Periodicidad o Desplazamiento

x o 360 x90 x180 x

sin(x) = sin xsin( 2 x) = + cos xsin( x) = + sin x

cos(x) = + cos xcos( 2 x) = + sin xcos( x) = cos x

tan(x) = tan xtan( 2 x) = + cot xtan( x) = tan x

csc(x) = csc xcsc( 2 x) = + sec xcsc( x) = + csc x

sec(x) = + sec xsec( 2 x) = + csc xsec( x) = sec x

cot(x) = cot xcot( 2 x) = + tan xcot( x) = cot x

x + 90x + 180x + 360

sin(x + 2 ) = + cos xsin(x + ) = sin xsin(x + 2) = + sin x

cos(x + 2 ) = sin xcos(x + ) = cos xcos(x + 2) = + cos x

tan(x + 2 ) = cot xtan(x + ) = + tan xtan(x + 2) = + tan x

csc(x + 2 ) = + sec xcsc(x + ) = csc xcsc(x + 2) = + csc x

sec(x + 2 ) = csc xsec(x + ) = sec xsec(x + 2) = + sec x

cot(x + 2 ) = tan xcot(x + ) = + cot xcot(x + 2) = + cot x

Calculo de Funciones Trigonometricas

FuncionDerivadaIntegral

sin xcos x cos x + C

cos x sin xsin x + C

tan xsec2 x = 1 + tan2 x ln |cos x| + C

csc x csc x cot x ln |csc x + cot x| + C

sec xsec x tan xln |sec x + tan x| + C

cot x csc2 x = (1 + cot2 x)ln |sin x| + C

Ley de Senos

a=b=c

sin Asin Bsin C

Ley de Cosenos

a2 = b2 + c2 2bc cos A b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C

Ley de Tangentes

tanA B

a b=2

a + btanA + B

2

tanB C

b c=2

b + ctanB + C

2

tanA C

a c=2

tanA + C

a + c

2

Identidades Pitagoricas

cos2 x + sin2 x = 1

sec2 x tan2 x = 1

csc2 x cot2 x = 1

2

sin x = 1 cos x

pcos x = 1 sin2 x

Suma y Diferencia de Angulos

sin(x y) = sin x cos y cos x sin y

cos(x y) = cos x cos y sin x sin y

tan(xy) =tan x tan y

1tan x tan y

1

csc(x y) =

sin(xy)

sec(x y) =1

cos(xy)

cot(xy) =cot x cot y 1

cot ycot x

Producto a Suma

cos x cos y = cos(x y) + cos(x + y) 2

sin x sin y = cos(x y) cos(x + y) 2

sin x cos y = sin(x + y) + sin(x y) 2cos x sin y = sin(x + y) sin(x y) 2

Suma a Producto

sin xsin y =2 sinx ycosx y

22

cos x + cos y =2 cosx + ycosx y

2

2

cos xcos y =2 sinx + ysinx y

2

2

tan xtan y =sin(x y)

cos x cos y

Identidades de Angulo Doble

2 tan xsin 2x = 1 + tan2 x = 2 sin x cos x

cos 2x = 1 tan2 x = cos2 x sin2 x 1 + tan2 x= 2 cos2 x 1 = 1 2 sin2 x

2 tan x tan 2x = 1 tan2 x

cot 2x = cot2 x 1 2 cot x

cot 3x =

tan 3x =Identidades de Angulo Triple

sin 3x = 3 cos2 x sin x sin3 x = 3 sin x 4 sin3 x

cos 3x = cos3 x 3 sin2 x cos x = 4 cos3 x 3 cos x

3 tan x tan3 x1 3 tan2 x3 cot x cot3 x1 3 cot2 xIdentidades de Angulo Medio

sin2 x cos2 x = 1 cos 4x 8

sin3 x cos3 x = 3 sin 2x sin 6x 32

sin4 x cos4 x = 3 4 cos 4x + cos 8x 128

sin5 x cos5 x = 10 sin 2x 5 sin 6x + sin 10x 512

Diferencia de Cuadrados a Producto

sin2(x) sin2(y) = sin(x + y) sin(x y) cos2(x) sin2(y) = cos(x + y) cos(x y)

sin x2

cos x2

tan x2

cot x2

tanx + y

2

= r1cos x

2

= r1+cos x

2

= r1 cos x= csc xcot x

1 + cos x

=sin x

1 + cos x

= r1 + cos x= csc x + cot x

1 cos x

=sin x=1 + cos x

1 cos xsin x

=sin x + sin y =cos x cos y

cos x + cos ysin xsin y

Composicion de Funciones

sin(arccos x) = 1 x2tan(arcsin x) = x1 x2

sin(arctan x) = x 1 + x21 x2tan(arccos x) =x

cos(arctan x) = 1 1 + x21 x2cot(arcsin x) = x cos(arcsin x) = 1 x2

Reduccion de Exponentes

sin2 x = 1 cos 2x 2

sin3 x = 3 sin x sin 3x 4

sin4 x = 3 4 cos 2x + cos 4x 8sin5 x = 10 sin x 5 sin 3x + sin 5x 16

cos2 x = 1 + cos 2x 2cos3 x = 3 cos x + cos 3x 4

cos4 x = 3 + 4 cos 2x + cos 4x 8

cos5 x = 10 cos x + 5 cos 3x + cos 5x 16

cot(arccos x) = x1 x2

Suma y Diferencia de Inversas

arcsin x + arccos x = 2 arctan x + arccot x = 2(1 ,si x > 0

arctan x + arctan=2

x

2 ,si x < 0

arcsin x arcsin y

p

= arcsin(x1 y2 y 1 x2)

arccos x arccos y

arctan xarccos(xy(1x2)(1y2))

= arctan yp

= arctany

x

1 xy