If is measured in radian

Then :

If is measured in radian

Then :

and:

< -1-cos <

< -sin <

From the theorem , and by sandwich theorem ,

We find that

a < < - )

sin

0 0 0As 0

lim0

sin =0

And b ) < -1-cos < so

0 0 0As 0

lim0 1-cos =0

lim0 Cos = 1

ExampleFind the limit if it exists:

0

sinlim

Example

g()=1

h()=cos

sin( )f

Example

0lim1 1

0

lim cos 1

&

therefore…

0

sinlim 1

sincos 1

If we graph , it appears thatsin x

yx

0

sinlim 1x

x

x

Find the following limits:

a)x

x3)3sin(lim

0x

Put = 3x so as x 0 , = 3x 0

1)sin(

=lim0xx

x3)3sin(= lim

0x

Solution

xx

5)3sin(b) lim

0x

= 53

xx

5)3sin(

=lim0x 5

3x

x3)3sin(

=lim0x 5

3x

x3)3sin(lim

0x

Solution

Put = sin x so as x 0 ,

= sin x 0

=

00

Find xx

sin)sin(sinlim

0x

xx

sin)sin(sinlim

0x

= = 1x

xsin

)sin(sinlim0x

)sin(lim0x

Solution

Find:)8sin()3tan(

xxlim

0x

=

=

=

=

11.1.3.1.

81

=83

=

00

)8sin()3tan(

xxlim

0x

)3cos(1.)3sin(.

)8sin( xxx

xx

)3cos(1).3sin(.

)8sin(1

xx

xlim0x

lim0x

)3cos(1.

3)3sin(.3.

)8sin(8.

81

xxx

xxlim

0x

Solution

1 )Find )2

(

)2

sin(

t

tlim2t

)2

(

)2

sin(

t

t=

00lim

2t

Put = t- so as t , 0. 2 2

= 1)2

(

)2

sin(

t

t=

lim2t

)sin(lim0

Solution

Derivative of y = sin xDerivative of y = sin x

sin cosd

x xdx

WHY?WHY?

Derivative of y = sin xDerivative of y = sin x

0( ) ( )

( ) limhdy f x h f x

f xdx h

0sin( ) sin( )

( ) limhx h x

f xh

0sin( )cos( ) cos( )sin( ) sin( )

limhx h x h x

h

0sin( )cos( ) sin( ) cos( )sin( )

limhx h x x h

h

0sin( )(cos( ) 1) cos( )sin( )

limhx h x h

h h

= 0 +cos(x)*1 = cos (x)

Derivative of Sine, CosineDerivative of Sine, Cosine

cos sind

x xdx

sin cosd

x xdx

Examples:

f(x) = x2 +sin(x)

f’(x) = 2x + cos(x)

•f(t) = cos(t) – 5t -2 , then f '(t) = -sin (t) +10t -3

Derivative of Sine, Cosine

Find tan x dxd

dxd tan x = dx

dxx

cossin

xxxxx

2cossin.sincos.cos =

2 2cos sin 12 2cos cos

x x

x x

=

Solution

Derivatives of Trigonometric Derivatives of Trigonometric FunctionsFunctions

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

Find if: dxdy

1 (Y= xxx secsin32

xxxxdxdy tanseccos32

Solution

2 )Y = xxx csc.tancos

)2secsin(csc)cotcsc).(tan(cos xxxxxxxdxdy

Find y’ if y = xx

sin1cos

Solution

21( ) 5sin sec tan 7 3

2f x x x x x x

21( ) 5cos sec tan sec tan (1) 14

2f x x x x x x x x

1 sin( )

cos

xf x

x x

2

( cos ) (1 sin ) (1 sin ) ( cos )( )

( cos )

d dx x x x x x

dx dxf xx x

2

( cos )(cos ) (1 sin )(1 sin )( )

( cos )

x x x x xf x

x x

2 2 2 2

2 2

( cos cos ) (1 sin ) cos cos 1 sin( )

( cos ) ( cos )

x x x x x x x xf x

x x x x

2

cos( )

( cos )

x xf x

x x

Find :

=

0tancos)0sec(2

= 13

= 3

xx

tancossec2

lim0x

xx

tancossec2

lim0x

Solution

Has a tangent at x = 0 .

Show that the function:

F)x)=

0

)1sin(.2 xx X 0

X = 0

Find the values of a and b s.t the function:

F)x) =

bax

x 12 ,x 2

,x < 2

Is diff . Every where.