Transcript Title

the Further Mathematics network
www.fmnetwork.org.uk
the Further Mathematics network
www.fmnetwork.org.uk
FP2 (MEI)
Hyperbolic functions Introduction (part 1)
Let Maths take you Further…
Introduction to hyperbolic functions
Before you start:



You need to be confident in manipulating exponential and logarithmic
functions
You need to be confident all the calculus techniques covered in Core 2 and
3
You need to have covered chapter 4 on Maclaurin series
When you have finished…
You should:


Understand the definitions of hyperbolic functions and be able to sketch
their graphs
Be able to differentiate and integrate hyperbolic functions
Exploring with Autograph
px  qy  1
2


2
What does the graph look like if p=q=1?
What happens if we change the values of
p & q (where p & q are real constants)?
Cartesian and parametric forms
Unit circle
x  y 1
2
2
Cartesian and parametric forms
Rectangular hyperbola
x  y 1
2
2
Difference of two squares:
1  1
x  t  
2 t 
1  1
y  t  
2 t 
let
t e
u
x
y
But notice
the
restriction
that now t>0
Compare!




1 i
 i
cos   e  e
2
1 i
sin  
e  e  i
2i




1 u
u
cosh u  e  e
2
1 u
u
sinh u  e  e
2
What do these hyperbolic
functions look like?

1 u
cosh u  e  e u
2

What do these hyperbolic
functions look like?

1 u
sinh u  e  e u
2

Cartesian and parametric forms
Rectangular hyperbola
x  y 1
2
2
x  cosh u
y  sinh u
These are not the
standard parametric
equations that are
generally used, can you
say why not?
x  sec 
y  tan
are used
Complex variables, z

1 z
cosh z  e  e  z
2
Replace z by iz


1 z
z
sinh z  e  e
2
Replace z by iz

Complex variables, z

1 iz
cos z  e  e iz
2
Replace z by iz


1 iz
sin z 
e  e iz
2i
Replace z by iz

Results
cosh(iz) = cos z
sinh(iz) = i sin z
cos(iz) = cosh z
sin(iz) = i sinh z
Circular trigonometric identities and
hyperbolic trigonometric identities
cos (iz)  sin (iz)  1
2
2
Osborn’s rule

“… change each trig ratio into the
comparative hyperbolic function, whenever a
product of two sines occurs, change the sign
of that term…”
cos   sin   1
2
2
cos   sin   cos2
2
2
Differentiation
Integration
Calculus - Reminder
The usual techniques can be used….
Calculus - Reminder
The usual techniques can be used…
Introduction to hyperbolic functions
When you have finished…
You should:


Understand the definitions of hyperbolic functions and be able to
sketch their graphs
Be able to differentiate and integrate hyperbolic functions
Independent study:
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
Using the MEI online resources complete the
study plan for Hyperbolic functions 1
Do the online multiple choice test for this and
submit your answers online.