Ch 1 Differentiating Trigonometric Functions

8/3/2019 Ch 1 Differentiating Trigonometric Functions

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CORE 4

DIFFERENTIATINGTRIGONOMETRIC

FUNCTIONS


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Standard Trigonometric Derivatives

y = sin x y = cos x y = tan x

dy = cos x dy = -sin x dy = sec2 xdx dx dx

y = sec x y = cosec x y = cot x

dy = sec x tan x dy = -cosec x cot x dy = -cosec2 xdx dx dx

NB Formulas in red need to be learned they are not in your formulabooklet.


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Example 1 using the chain rule

a) y = sin (3x /4) b) y = cos4 x


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Example 2 using the product rule

a) y = x3 sin x b) y = 1 cos xx


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Example 3 using the quotient rule

Prove d ( tan x ) = sec2 xdx


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Example 4 combining rules

a) y = sin2 3x


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Example 4 combining rules

b) y = sin5 x cos3 x


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Applications of differentiating trigonometric functions

Example 5The height in metres of the water in a harbour is given approximatelyby the formula h = 6 + 3cos t where t is measured

6in hours from noon. Find an expression for the rate at which the wateris rising at time t. When is it rising fastest?


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Example 6Find the maxima and minima of f(x) = 4 cos x + cos 2x


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Example 7Find the points on the graph of y=x sin x at which the tangent passesthrough the origin


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Standard Trigonometric Integrals

sin x dx = - cos x + c cos x dx = sin x + c

tan x dx = ln |sec x | + c

sec

2

x dx = tan x + c sec xtan x dx = sec x +c

Using chain rule

cos ax dx = 1 sin ax + ca

NB Formulas in red need to be learned they are not in yourformula booklet.


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Example 8

a) tan (2x ) dx b) tan2 x dx4


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Note

When integrating squared trigonometric functions,use trigonometric identities to substitute toremove powers

Common identities to use are:

cos2 x = + cos 2x

sin2 x = - cos 2x

sinx cos x = sin 2x


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Example 9

Find the area under the graph of y = sin (2x + ) from x = 0 as far3

as the first point at which the graph cuts the x axis.


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Example 10

Let R be the region under the graph of y = sin2 x in the interval0x. Find(a) the area of R(b) the volume of revolution formed by rotating R about the x axis


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Homework

Page 219 Exercise 1C

Questions 2adg, 4ac, 5ac, 9, 12

Ch 1 Differentiating Trigonometric Functions

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Transcript of Ch 1 Differentiating Trigonometric Functions