7: Differentiating some 7: Differentiating some Trig FunctionsTrig Functions

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Differentiating some Trig Functions

A reminder of the rules for differentiation developed so far!

1 nn nxdx

dyxy

xx edx

dyey

The chain rule ( for functions of a function ):

dx

du

du

dy

dx

dyxgfy ))((

where )(xgu

( I call these functions the simple ones. )

Differentiating some Trig FunctionsThe trig functions are quite different in shape

from either of the simple functions we’ve met so far, so the gradient functions won’t follow the same rules.We’ll start with and use degrees

xy sinxy sin

x

y

We only need the 1st quadrant as symmetry will then give us the rest of the gradient function.

Differentiating some Trig Functions

dx

dy

x

x

y xy sin

x

x

xx

The gradient function x

x

x

x

The function

Differentiating some Trig Functions

dx

dy

x

x

y xy sin

x

x

xx

The gradient function x

x

x

The gradient drops more slowly between and . . .

0x 30x90x60xthan between and .

The function

x

Differentiating some Trig Functions

dx

dy

x

x

y xy sin

x

x

xx

x

x

x

x

The gradient function

The function

Differentiating some Trig Functions

x

y xy sin

x

x

x

xx

x

x

x

x dx

dyThe gradient function

The function

Differentiating some Trig Functionsxy sin

x

y

x

x

x

x

dx

dyx

The function

We can estimate the gradient at x = 0 by using the tangent.

1

60

So, the gradient is 60

1

60

1

dx

dy

The gradient function looks like BUTwe need a scale on

xcos

the axis.

Differentiating some Trig Functions

x

x

x

x

dx

dyx

xy sin

x

y

The function1

60

60

1

xcos

The gradient function isn’t since not .

10cos 601

xdx

dyxy cos

60

1sin

So,

Differentiating some Trig Functions

xy sin

x

y

However, if we use radians:

1

5712

1

dx

dy

x

x x

x

1

It can be shown that this length . . . is exactly 1.

xdx

dycos

xy sinSo,

xy cos

Differentiating some Trig Functions

From now on we will assume that x is in radians unless we are told otherwise.

xdx

dyxy cossin We

have,

Exercise

Using radians sketch for . xy cos 2

2 x

Underneath the sketch, sketch the gradient function. Suggest an equation for the gradient graph.

Differentiating some Trig FunctionsSolution:

xdx

dysin

xdx

dy

xy cosx

yxy cos

This is a reflection of in the x-axis so its equation is

xsin

Differentiating some Trig FunctionsSUMMAR

Y If x is in radians,

xdx

dyxy cossin

xdx

dyxy sincos

We need a bit more theory before we can differentiate the trig function . This is done in a later presentation.

xy tan

N.B. We have not proved these results; just shown they look correct.

Differentiating some Trig FunctionsCompound Trig

FunctionsWe can use the chain rule to differentiate trig functions of a function.

Solution: (a) First find the gradient function:

uy sin

3dx

duxu

du

dy3coscos

dx

du

du

dy

dx

dy

e.g. 1 Find the gradient of at the point where .

xy 3sinx

xu 3Let

3cos3dx

dyx 3)1(3

N.B. We don’t need to put brackets round 3x.

N.B. Radians!

x3cos3dx

dy

Differentiating some Trig Functions

xcos2

e.g. 2 Differentiate xy 2cosWhat would you let u equal in this example?

xu cos 2uy

xdx

dusin u

du

dy2

)sin(cos2 xx dx

du

du

dy

dx

dy

If we write as we can easily see that the inner function is .

x2cos 2)(cos xxcos

So, let

xxdx

dysincos2

Solution:

dx

dy

Differentiating some Trig FunctionsExercise

Differentiate the following with respect to x:

xy 3cos xy sin2

xy 2sin

1. 2.

Solutions:

xy 3cos1.

3dx

duxu

du

dy3sinsin

dx

du

du

dy

dx

dy x

dx

dy3sin3

xu 3 uy cosLet

Differentiating some Trig Functions

xy sin22.

xdx

ducos xu

du

dysin22

dx

du

du

dy

dx

dy xx

dx

dycossin2

xu sin 2uy Let

xdx

dycos2

xy 2sin3. 2)(sin x

Differentiating some Trig Functions

Differentiating some Trig Functions

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Differentiating some Trig Functions

A reminder of the rules for differentiation developed so far!

1 nn nxdx

dyxy

xx edx

dyey

The chain rule ( for functions of a function ):

dx

du

du

dy

dx

dyxgfy ))((

where )(xgu

( I call these functions the simple ones. )

Differentiating some Trig FunctionsSUMMAR

Y If x is in radians,

xdx

dyxy cossin

xdx

dyxy sincos

Differentiating some Trig FunctionsCompound Trig

FunctionsWe can use the chain rule to differentiate trig functions of a function.

Solution: (a) First find the gradient function:

uy sin

3dx

duxu

du

dy3coscos

xdx

dy3cos3

dx

du

du

dy

dx

dy

e.g. 1 Find the gradient of at the point where .

xy 3sinx

xu 3Let

3cos3dx

dyx 3)1(3

N.B. We don’t need to put brackets round 3x.

N.B. Radians!

Differentiating some Trig Functions

xcos2

e.g. 2 Differentiate xy 2cos

xu cos 2uy

xdx

dusin u

du

dy2

)sin(cos2 xxdx

dy

dx

du

du

dy

dx

dy

If we write as we can easily see that the inner function is .

x2cos 2)(cos xxcos

So, let

xxdx

dysincos2

Solution: