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![Page 1: ACS123 Functions Dr Viktor Fedun Automatic Control and Systems Engineering, C09 Based on lectures by Dr Anthony Rossiter.](https://reader036.fdocuments.in/reader036/viewer/2022062404/551ae7fb5503466b6a8b5f28/html5/thumbnails/1.jpg)
ACS123Functions
Dr Viktor Fedun
Automatic Control and Systems Engineering, C09
Based on lectures by Dr Anthony Rossiter
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Why is mathematics important?
Why do engineers need to be good at mathematics?
Is it sufficient to memorise key results?
Just because a learning technique worked at school, does that make it the best method now?
What mathematics do I need to be good at?
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Mathematics is a tool-kit
A good engineer:
1. knows which is the best tool to use?
2. Is proficient in using the tool?
3. Can adapt the tool to a new use.
It is not good enough to memorise key results as the most important skill is abstraction. You must put your effort into understanding.
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Module assessment
• 3 in class tests in weeks 4, 7, 11, 13
These will be similar to exam questions.• An exam in May or June
If you want feedback on an answer you have done, ask in a tutorial.
20 credit module, similar pattern in semester 2.
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Module organisation
I will teach the first semester
Lectures and tutorials
Of the 5 timetabled hours, 2-3 will be used for lectures (these times may vary each week).
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MOLE
Please use the discussions board to ask questions. Then everyone can see the question and answer.
I will not respond to email queries unless of a personal or private nature.
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Resources
Learning is only effective where students engage in self-discovery.
1. What you hear, you will usually forget.2. You only really understand something when
you use it.3. We will provide ample materials, but YOU will
only learn if you use these properly. [5-6 hours per week]
Lecturers are here to guide – NOT TO TEACH! We will answer queries and be as helpful as possible, but only you can do the work.
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Be a function
Stand up. 1. Use your arms to illustrate y=x.2. What about y=-x?3. Can you do y=x2, or even x3.4. What about sine(x) – you may need a
partner. Now do cosine(x).5. Can you y=mod(x)? Or even y=sqrt(x2)?6. Can you think of any more?
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Common functions
1. sine, cosine, tangent (and their inverses)
2. logarithm, exponential
3. sinh and cosh
4. straightline, quadratic, general polynomial
5. combinations of above as products, composites and fractions.
You should be familiar with shapes of common functions and be able to sketch quickly.
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Example 1 (Page 136, Kuldeep and Singh, Example 3)
[Mechanics]
The displacement, φ(t), of a particle at time t is given by:
φ(t)= 2t3 + t2 - 10t + 10
– Evaluate φ(2), φ(3), φ(5).– Find simplified expressions for:
(i) φ(t2)
(ii) φ(t + 1)
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Example 1 Solution
Solution:
(a) We have
(a)φ(2) = (2 x 23) + 22 – (10 x 2) + 10 = 10
(a)φ(3) = (2 x 33) + 32 – (10 x 3) + 10 = 43
(b)φ(5) = (2 x 53) + 52 – (10 x 5) + 10 = 235
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Example 1 Solution
(b) (i) For φ(t2) we replace the t with t2 in φ(t)= 2t3 +
t2 - 10t + 10:
φ(t2) = 2(t2)3 + (t2)2 – 10(t2) + 10
= 2t6 + t4 – 10t2 + 10
(ii) For φ(t + 1) we replace t with t+1 in φ(t)= 2t3 + t2 - 10t + 10:
φ(t+1)= 2(t+1)3 + (t+1)2 – 10(t+1) + 10:
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What is a function?
1. A rule which translates an input, usually to a single output.
2. What are the functions for:i. Double the input
ii. Shift the input by 3
iii. Cube the input and subtract 1.
3. Write down in words the functions for
2)1(;4;25 xyxyxy
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What variables can a function have?
What is the difference between the functions f(x), g(w), h(y) and k(x)
A function describes a relationship, the variable names are unimportant.
Engineers typically use variable names that relate to the topic: W for weight, h for height, L for length, etc.
22 )2()()2()(
)sin()()sin()(
xxkwyyhx
wwgzxxfy
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What is a function argument?
The part that appears in the brackets;
• For y=f(x), x is the argument.
• For z=g(w), w is the argument.
Thus argument is another word for the input to the function.
Independent and dependent variables: what do you think these are? Use common sense.
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Composition of functions
What do the following statements mean?
;3)()));(((
)sin()(;)());((
)sin()(;)());((2
2
xxhzfghw
xxgxxfxfgy
xxgxxfxgfy
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Evaluate the following
Find y when x=pi/2.
Find w when z=1.
;3)()));(((
)sin()(;)());(( 2
xxhzfghw
xxgxxfxgfy
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Function products
Evaluate A given that:A = y2h with x=2 and z=3
Write down a detailed function expression to express A.
;3)()));(((
)sin()(;)());(( 2
xxhzfghw
xxgxxfxgfy
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Example 2 (Page 152 Kuldeep Singh, Example 16)
[Reliability Engineering]The failure density function, f(t), for a component is given by:
f(t) = 1/8 where 0 < t < 8 years.
Find F(t), R(t) and h(t) where these are defined as:F(t) = tf(t) (Failure Distribution function)R(t) = 1-F(t) (Reliability function)h(t) = f(t) / R(t) (hazard Rate function)
and 0 < t < 8 years.
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Example 2 (Page 152 Example 16)
SolutionWe have:
F(t) = tf(t) = t(1/8) = t/8.
R(t) = 1-F(t) = 1- t/8
h(t) = f(t) / R(t) = (1/8)/(1-t/8)
= 1/(8-t)
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Graphs and sketching
By first producing a suitable table, sketch the graphs of the following functions in the domain -3 to 3.
)tan(
)2)(1(
)2
sin(
2)sin(
xy
xxy
xy
xy
Domain is the
values allowed to the argument or independent variable.
Range is the values the output (dependent variable) can take. What is the range of these?
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Example 3 (Page 110 Example 7)
[Fluid Mechanics]
The streamlines of fluid flow are given by:
y = x2 + c
where c is constant.
Sketch the streamlines for c = 0, -1 ,1, -2, 2, -3 and 3.
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Example 3 (Page 110 Example 7)
Solution
The graphs of y = x2 + c for c = 0, -1 ,1, -2, 2, -3 and 3 are:
(c=0) y = x2 (c=-1) y = x2 - 1 (c=1) y = x2 + 1(c=-2) y = x2 -2(c=2) y = x2 +2(c=-3) y = x2 – 3(c=3) y = x2 + 3
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y
x
-3-2-10
123
c = -3
c = -2c = -1c = 0c = 1c = 2c = 3
-1-2 1 2
Notice how the graph of y=x2 + c varies as c changes. The c is where the curve cuts the y axis.
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Inverse functionmean that all we’ve done is made a switch in
emphasis
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Inverse functionmean that all we’ve done is made a switch in
emphasis
7 – 4 = 3 3 + 4 = 7
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Inverse functionmean that all we’ve done is made a switch in
emphasis
7 – 4 = 3 3 + 4 = 7
Both of this statements say the same thing, but with a change in emphasis
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Inverse functionmean that all we’ve done is made a switch in
emphasis
7 – 4 = 3 3 + 4 = 7
Both of this statements say the same thing, but with a change in emphasis
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Inverse functionmean that all we’ve done is made a switch in
emphasis
7 – 4 = 3 3 + 4 = 7
Both of this statements say the same thing, but with a change in emphasis
-1
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Inverse functionexample
y=2x-7; y=f(x)=2x-7
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Inverse functionexample
y=2x-7; y=f(x)=2x-7
Identity function
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Inverse functionexample
y=2x-7; y=f(x)=2x-7
Identity function
If and inverse function
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Inverse functionexample
y=2x-7; y=f(x)=2x-7
Identity function
If and inverse function
Composition of functions
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Inverse function
A function f and its inverse f . Because f maps 1 to 4,
the inverse f maps 4 back to 1.
-1
-1
-1 f f
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One to one function
For every value of x, there is a distinct value of y and for every value of y there is a distinct value of x.
Which of the following is one to one?
)sin(
13
242
xy
eyxxy
xy
x
Draw the graph and it should be obvious.
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Inverse function
What about?
3
223
)(sin)sin( 1
yxxy
yxxy
)(cos)sin( 12 xxxy
Proof
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Inverse functionexample
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Inverse functionexample
Sometimes the inverse of a function cannot be expressed by a formula
with a finite number of terms. For example, if f is the function
then f is one-to-one, and therefore possesses an inverse function f . The formula for this inverse has an infinite number of terms:
-1
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Inverse functionexamples
link
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Many-to-one and one-to-many
Give some examples of many-to-one and one-to-many functions.
The logic goes from independent variable to dependent variable.
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Notation
Get into groups and decide three example functions with the following properties [3 for each item].
1. Continuous
2. Discontinuous
3. Periodic (Why are these important?)
4. Odd
5. Even
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Odd Even
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Summary
Independent variable (domain)Dependent variable (range)FunctionMany-to-one (one-to-one,…)Odd, even, periodicInverse functionContinuous/discontinuousComposite functionStraight lines
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Exponential functions
On some rough paper, do a sketch of the following functions.
In what sense are the functions equivalent?
23
2
1
5
3
2
x
x
x
y
y
y
With a suitable rescaling of x, they are all the same shape.
Functions of this form are called exponentials.
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-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
9
y1
y2y3
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Exponential properties
If you double the value of the independent variable, you square the value of the dependent variable.
There is a constant ratio which depends solely on the difference of the argument:
3
2
)]([)3(
)]([)2(
xfxf
xfxf
)]([)(
)(xbaf
bxf
axf
For all x!
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Exponential properties
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Exponential properties
Exponentiation is not commutative
4 + 5 = 5 + 4 4 * 5 = 5 * 4 but 4 = 55 4
256 = 625
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Exponential properties
Exponentiation is not commutative
4 + 5 = 5 + 4 4 * 5 = 5 * 4
but
4 = 55 4
256 = 625Exponentiation is not associative
(2 + 3) + 4 = 2 + (3 + 4) (2 * 3) * 4 = 2 * (3 * 4)
but
2 = 40963( )
42 = 2.417.851.639.229.258.349.412.353
3( )4
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Exponential convention
1. When dealing with exponential functions it is usual to assume the same base – ALWAYS!
2. The assumed base is `e’.
3. It will become clearer later why `e’ is chosen because this makes a lot of common algebra much simpler.
4. `e’ is irrational, but has a value near 2.7
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Exponential convention
1. When dealing with exponential functions it is usual to assume the same base – ALWAYS!
2. The assumed base is `e’.
3. It will become clearer later why `e’ is chosen because this makes a lot of common algebra much simpler.
4. `e’ is irrational, but has a value near 2.7and more precisely
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Common exponential
The most common functions you will deal with are:
A positive exponent gives an increasing function with increasing argument.
A negative exponent gives a decreasing argument with exponent.
bt
at
x
x
ety
ety
xey
xey
)(
)(
)exp(
)exp(
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-1 -0.5 0 0.5 1 1.5 2 2.5 30.5
1
1.5
2
exp(0.2t)exp(-0.2t)
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Exponentials and systems engineering
The behaviour of real systems is often described as an exponential decay.
1. Radioactivity follows a curve of the form.
2. An explosion might be an increasing function.
3. Many systems have dynamics with 2 exponentials.
ctehth )0()( btat BeAetz )(
atertr )0()(
If behaviour has a positive exponent – BEWARE!
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Note
Logarithms and exponentials are inverse functions of one another.
1log
)(log3
log)(log)()exp()(
log
log
3
1
e
AeorAe
zwez
xxfxxf
yxey
e
AAe
ew
e
ex
e
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Even/odd and hyperbolic Functions
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Even functions
An even function is one whereby the vertical axis is equivalent to a mirror.
In mathematical terms, this means that
f(-x)=f(x)
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Examples of even functions
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x2-2cos(x)
sin(x)2
Notice symmetry about x=0
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Odd functions
An odd function is one whereby the vertical axis reverses the value of the function.
In mathematical terms, this means that
f(-x)=-f(x)
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Examples of odd functionsNotice asymmetry about x=0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-6
-4
-2
0
2
4
6
x3-2xsin(x)tan(x)
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Constructing even and odd functions
Every function can be made up of even and odd functions. This can make some engineering problems easier to handle.
EVEN FUNCTIONS
f(x)=f(-x)
ODD FUNCTIONS
f(x)=-f(-x)
)]()([2
1)]()([
2
1)(
)]()([2
1)]()([
2
1)(
xfxfxfxfxf
xfxfxfxfxf
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Constructing even and odd functions
EVEN ODD
EVEN FUNCTIONS
f(x)=f(-x)
ODD FUNCTIONS
f(x)=-f(-x)
)()]()([2
1)(
)()]()([2
1)(
)]()([2
1)]()([
2
1)(
xhxfxfxh
xgxfxfxg
xfxfxfxfxf
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Construct even and odd functions to make up the following
wewwwg
zzh
xxxf
2tan)(
)3
sin()(
)cos(3)(
2
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Construct even and odd functions to make up the following
22tan
22tan
sin3
coscos3
sin)3
sin(
3)cos()cos(3
22wwww
w eeww
eeeww
zzz
xxxx
)]()([2
1
)]()([2
1
xfxfODD
xfxfEVEN
USE
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Simple rules
• EVEN*EVEN = EVEN
• ODD*ODD = EVEN
• EVEN*ODD=ODD
• EVEN+EVEN=EVEN
• ODD+ODD=ODD
• ODD+EVEN=NEITHER ODD NOR EVEN
Can you prove these?
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Common even/odd functions
EVEN
cos
x2n
cosh
ODD
sin
x2n+1
tan
sinh
Today we focus on cosh and sinh
2sinh;
2cosh
xxxx eex
eex
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Plots of cosh and sinh
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-4
-3
-2
-1
0
1
2
3
4
cosh(x)sinh(x)
2sinh;
2cosh
xxxx eex
eex
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Engineering examples of cosh and sinh
• Some examples taken from the following book:– “Engineering Mathematics through
Applications”• Kuldeep Singh
– Published by: Palgrave MacMillan– ISBN 0-333-92224-7
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Example 1 (Page 251 Example 21)
[Electrical Principles]
A transmission line of length L has voltage V. At a distance x from the sending end, the voltage is given by:
VL = ½(V + IZ0)e-φx + ½(V - IZ0)eφx (*)
Where I is the current, Z0 is the characteristic impedance and φ is the propagation coefficient.
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Show that at x = L:
VL = Vcosh(φL) - IZ0sinh(φL)
VL = ½(V + IZ0)e-φx + ½(V - IZ0)eφx (*)
Solution:
Putting x = L into (*) gives:
VL = ½(V + IZ0)e-φL + ½(V - IZ0)eφL
= V(e-φL + eφL)/2 + IZ0(e-φL - eφL)/2
= Vcosh(φL) + IZ0(e-φL - eφL)/2
so VL = Vcosh(φL) + IZ0sinh(φL)
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Example 2 (Page 251 Example 22)
[Electronics]
In a semiconductor, a force, F, exerted on an electron is given by:
F = Qcke-kx/(1+e-kx)2 (*)
Where c and k are constants, x is the distance from the pn junction and Q is the charge.
Show that
F = Qck/2[1+cosh(kx)]
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Solution:
F = Qcke-kx/(1+e-kx)2 (*)
so F = Qcke-kx/(1 + 2e-kx + e-2kx) = Qcke-kx / e-kx (ekx + 2 + e-kx) = Qck /(ekx + 2 + e-kx) = Qck /(2 + ekx + e-kx) = Qck /(2 + 2(ekx + e-kx)/2) = Qck /(2 + 2cosh(kx))
hence F = Qck/2[1+cosh(kx)]
Show that F = Qck/2[1+cosh(kx)]
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Solution to ODES
Where an ODE takes the form
The solution can be represented in two similar ways.
022
2
xadt
xd
atDatCxORBeAex atat sinhcosh;
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Identities
You should be familiar with common identities using cosh and sinh.
Prove the following:
xxxxxxx
xyyxyxyxyxyx
xx
coshsinh22sinh1cosh2sinhcosh2cosh
coshsinhcoshsinh)sinh(sinhsinhcoshcosh)cosh(
1sinhcosh
222
22
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Use of hyperbolic equations with parametric descriptions
Some simple curves lend themselves to parametric descriptions. Consider:
1. Circle
2. Ellipse
3. Hyperbola 222
22222
222
ryx
rybxa
ryx
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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
222 1 yx
22
2
2
2
16.02
yx
222 1 yx
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Parametric descriptions
sin,cos1222 yxyx
sin6.0,cos216.02
22
2
2
2
yxyx
tytxyx sinh,cosh1222
ENGINEERING APPLICATION: Space orbits can be either elliptical or hyperbolic (often called a sling shot).
the sine and cosine functions give a parametric equation for the ellipse
the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola
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Questions
1. Simplify the following expressions.
2. Find parametric expressions for x,y satisfying the following hyperbola.
)sinh()cosh(
)2sinh2(cosh2cosh3
2cosh4sinh5.0 2
yxyxh
fxxgx
xxf
2162 22 yx
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Link between cosine, sine, cosh and sinh
You may find the following useful.
ixee
xi
ixee
x
xixe
ixix
ixix
ix
sinh2
)sin(
cosh2
)cos(
sincos