SCHOOL OF MATHEMATICAL SCIENCES
ENG1091
Mathematics for Engineering
Assignment 4: Hyperbolic Functions
Due date: Lab class for week 7
There are two questions in this assignment worth a total of 30 marks (5% of the final
mark for this unit).
The first question (15 marks) is an exam style question.
The second question (15 marks) is a traditional assignment style question.
The questions will be marked using distinct criteria as detailed in the unit-guide and in
the examples on the unit website. Please read these carefully.
For full marks you will need to show all of your working.
Late submissions will be subject to late penalties (see the unit guide for full
details).
Short answer exam style question
1. (a) Starting with the equation
x = tanh y
show that
−1
y = tanh
1
x = loge
2
1+x
1−x
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(b) Compute the following derivative
d
sinh−1 (cos(x))
dx
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(c) Assume θ is a real number. Then use Euler’s formula eiθ = cos θ + i sin θ to show
that
sinh(iθ) = i sin(θ)
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School of Mathematical Sciences
Monash University
(d) Use the definitions
cosh(x) =
1 x
e + e−x ,
2
sinh(x) =
1 x
e − e−x
2
to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x).
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Detailed answer assignment style question
2. Most of us will have no problem in solving simple linear equations like
0 = 3x + 7 ,
x =?
or even a slightly tricker quadratic equations such as
0 = 4x2 − 2x − 7 ,
x =?
This is bread-and-butter stuff for secondary school students. But you are bright and
enthusiastic university maths students looking for a challenge. Bring it on. Indeed. So
why not solve this cubic equation
0 = 4x3 + 3x − 2 ,
x =?
Should be a piece of cake? Not so fast young grasshopper. As you will soon see the
answer involves hyperbolic functions (surprised?). So let the games begin.
(a) Show, using the standard definitions of sinh(y), that
0 = 4 sinh3 y + 3 sinh y − sinh 3y
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(b) Use the result of part (a) to show that a solution to
0 = 4x3 + 3x − 2
is
x = sinh y = sinh
1
−1
sinh 2
3
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(c) Generalise this result to cubics of the form
0 = ax3 + bx + c
where a and b are positive real numbers. (Hint: try re-scaling x to produce a new
cubic of the form used in part (b)).
26-Jul-2014
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