32 )1( xDifferentiate: What rule are you using?

Aims:• To learn results of trig differentials for cos x, sin x & tan x.• To be able to solve trig differentials that require the chain, product and quotient rules • To be able to reverse this process in order to integrate definite and indefinite expressions.

Trigonometry Lesson 5 (Differentiation & Integration)

The derivative of sin x

The derivative of sin xBy plotting the gradient function of y = sin x, we deduce that

dyy x xdx

If = sin then = cos

Functions of the form k sin f(x) can be differentiated using the chain rule.

Differentiate y = 2 sin 3x with respect to x.

So if y = 2 sin u where u = 3x

Using the chain rule:

In general using the chain rule, f f fIf = sin ( ) then = '( )cos ( )dyy x x x

dx

The derivative of cos x

The derivative of cos xBy plotting the gradient function of y = cos x, we deduce that

dyy x xdx

If = cos then = sin

Differentiate y = 3 cos (x3 + 4) with respect to x.

And similarly to the differentiation of sine, it can be shown using the chain rule,

f f fIf = cos ( ) then = '( )sin ( )dyy x x xdx

Differentiate these!11. sin4

2. cos(3 2)

x

x x

x

3

5

cos2 .4

sin .35(sinx) as same theiswhich

Try these on w/b’s1. sin 9

62. -cos7

x

x

4

5

3. 3sin

4. cos 6

x

x

4which is the same as (sinx)

The derivative of cos x

Find given that y = –x2 cos x.dydx

Let u = and v =

So

Using the product rule:

We can also now differentiate products and quotients which contain trig functions.

The derivative of tan xWe can differentiate y = tan x by writing it as

xyx

sin= cos

Then we apply the quotient rule with u = sin x and v = cos x :

dyy x xdx

2If = tan then = sec

dyy x xdx

= sin = cos

Derivatives & Integrals of trigonometric functionsThese are the differential we have seen:

dyy x xdx

= cos = sin

dyy x xdx

2= tan = sec

As integration is the reverse of differentiation it follows that:

cos sinx dx x c

sin cosx dx x c 2sec tanx dx x c

In A2 you will be only asked to integrate sine and cosine functions. It can also be shown using the reverse chain rule that when dealing with the cos and sin of linear functions:

1cos( + ) = sin( + ) +ax b dx ax b ca

1sin( + ) = cos( + )+ax b dx ax b ca

Trig Integration

Find 2sin 5x dx

Find 2

0(2 cos 4 )x dx

Complete wheel puzzle Exercise F pg 101 qu 1, 3, 5, 7. Exercise G pg 104 qu 1 c, f, e. 2 and 3. Then Exercise A pg 108 qu 2, 3b, 4b,d,f and 5b, c, f

(b)

Person A

Person B