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
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)?
122 qypx
Cartesian and parametric forms
122 yxUnit circle
Cartesian and parametric forms
122 yxRectangular hyperbola
Difference of two squares:
ttx1
2
1
tty1
2
1
uet let
But notice the restriction that now t>0
y
x
Compare!
ii ee 2
1cos
ii eei
2
1sin
uu eeu 2
1cosh
uu eeu 2
1sinh
What do these hyperbolicfunctions look like?
uu eeu 2
1cosh
What do these hyperbolic functions look like?
uu eeu 2
1sinh
Cartesian and parametric forms
122 yxRectangular hyperbola
ux coshuy sinh
These are not the standard parametric equations that are generally used, can you say why not?
secxtany
are used
Complex variables, z
zz eez 2
1cosh zz eez
2
1sinh
Replace z by iz Replace z by iz
Complex variables, z
iziz eez 2
1cos iziz ee
iz
2
1sin
Replace z by iz 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 1)(sin)(cos 22 iziz
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…”
1sincos 22 2cossincos 22
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:
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.