13 Inverse Trig Functions.ppt

7/27/2019 13 Inverse Trig Functions.ppt

http://slidepdf.com/reader/full/13-inverse-trig-functionsppt 1/19

13: Inverse Trig Functions

© Christine Crisp

“Teach A Level Maths”

Vol. 2: A2 Core Modules



Inverse Trig Functions

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with

permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Module C3

http://localhost/var/www/apps/conversion/tmp/scratch_3/0%20Contents.ppt




,

50sin x

If we use a calculator to find x for

we use the inverse sin function, .)50(sin1

Our calculators give us one solution, the principal

value, although we know there are an infinite numberof solutions.

We will now look in detail at the inverse functions:

x x x 111

tancossin

, ,





The notation for the inverse of is)(x f )(1

x f

A Bit More on Notation

WARNING The notation for a reciprocal and aninverse are the same!

x 11x e.g. means the reciprocal of x ,

50sin1

(it means undo the sine function )

but means the inverse of x sin

For the reciprocal of we must write .x sin 1sin

x





x y sin

What domain would you use?

We need to make sure that we have all the y -valueswithout any being repeated.

9090 x ( or, in degrees, as ).

The domain is defined as22

x





x y sin

To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on both axes.

5712

571

2





To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on the axes.

5712

x y sin

x y





To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on the axes.

5712

x y sin

x y x y 1

sin





x y 1

sin

The domain of an inverse function, , has the

same values as the range of and vice versa.

)(1

x f

)(x f 11 x So, the domain of isx y

1sin

and the range is

22

y

Domain and Range

1 1

2

2





x y 1sin

Finally, we can redraw using labels onthe axes that clearly show the domain and range.

x y 1

sin

11 x

22

y

It’s important to notice ( for later ) that thegradient is always positive except at whereit is undefined.

1x





Exercise

1. (a) Sketch for a suitable domain inorder to define its inverse.

(b) Sketch the inverse.

(c) Write down the domain and range.

(d) Write down what you notice about the gradient.

x y 1cos

x y 1

tan2. Repeat question 1. for .





x y 1

tan2

y

2

y

x y tan

x y

x y tan

x y

x y 1

tan

Solution: 2 or, using the usual scales,

(c) The domain of isx y 1

tan

and the range is

22

y

This is a strict inequality because of the asymptotes

x





The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and

photocopied. For most purposes the slides can be printedas “Handouts” with up to 6 slides per sheet.





11 x 22

y

It’s important to notice ( for later ) that thegradient is always positive except at whereit is undefined.

1x

Domain:

x y 1

sin

Range:




11 x y 0

The gradient is always negative except atwhere it is undefined.

1x

Domain: Range:

x y 1

cos




22

y x

The gradient is always 0

Domain: Range:

x y 1tan

13 Inverse Trig Functions.ppt

Documents

Transcript of 13 Inverse Trig Functions.ppt