FORMULARIO

FORMULARIO

TRIGONOMETRIA

sin2 x + cos2 x = 1;

tan x =

sin x

cos x ;

coth x =

cos x

sin x

sin(?x) = ? sin x; cos(?x) = cos x; sin( ��2 �� x) = cos x; cos( ��2 �� x) = ? sin x;

sin(�� �� x) = ? sin x; cos(�� �� x) = ? cos x; sin(x + 2��) = sin x; cos(x + 2��) = cos x;

sin(x �� y) = sin x cos y �� cos x sin y;

cos(x �� y) = cos x cos y ? sin x sin y

sin(2x) = 2 sin x cos x; cos(2x) = cos2 x ? sin2 x = 2 cos2 x ? 1 = 1 ? 2 sin2 x

cos2 x = 1+cos(2x)

; sin2 x = 1?cos(2x)

2

2

u?v

sin u + sin v = 2 sin u+v

2 cos 2 ;

u+v

u?v

cos u ? cos v = ?2 sin 2 sin 2 ;

u?v

sin u ? sin v = 2 cos u+v

2 sin 2 ;

sin x cos y = 12 [sin(x + y) + sin(x ? y)];

sin x sin y = ? 21 [cos(x + y) ? cos(x ? y)]

Posto t = tan(x/2), si ha:

sin x =

cos x cos y = 12 [cos(x + y) + cos(x ? y)];

2t

1+t2 ;

1?t2

1+t2 ;

cos x =

��

cos ��6 = 23 ;

sin 0 = 0

cos 0 = 1

sin ��6 = 12 ;

��

sin ��3 = 12 ;

sin ��2 = 1;

cos ��2 = 0;

sin ��3 = 23 ;

DISUGUAGLIANZE

|sin x| �� |x| per ogni x �� R;

SVILUPPI DI MACLAURIN

3

2

ex = 1 + x + x2! + x3! + �� �� �� +

log(1 + x) = x ?

x2

2

+

x3

3

x2

2

0 �� 1 ? cos x ��

log(1 + x) �� x per ogni x > ?1;

xn

n!

|xy| ��

x2 +y 2

;

2

(x+y)2

2

��

2

2 ;

cos ��4 =

n

x5

5!

x

+ �� �� �� + (?1)n (2n+1)!

+ o(x2n+2 )

2n+1

cos x = 1 ?

x2

2!

+

x4

4!

+ �� �� �� + (?1)n x2n! + o(x2n+1 )

2n

2 5

15 x

+

(1 + x)�� = 1 + ��x +

x5

5

+ o(x6 )

2n+1

+ �� �� �� + (?1)n x2n+1 + o(x2n+2 )

��(��?1) 2

x

2!

+

��(��?1)(��?2)) 3

x

3!

+ ������ +

1

��

2

2 ;

�� x2 + y 2 ; x4 + y 4 �� (x2 + y 2 )2

+ �� �� �� + (?1)n+1 xn + o(xn )

+

x3

3

sin ��4 =

2t

1?t2 ;

+ o(xn )

x3

3!

arctan x = x ?

tan x =

per ogni x �� R;

sin x = x ?

tan x = x + 13 x3 +

u?v

cos u + cos v = 2 cos u+v

2 cos 2 ;

��(��?1)������(��?n+1) n

x

n!

+ o(xn )

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