The expansion of sine and cosine in trigonometry by Maclaurin series. Expansion of the exponential function ex. Similarity found.

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THE TRIGO COMPANIONS
PM [B09]

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The ordinary vector
Let us come back to the oldie
vector to review what we have
started with.
It is the nice and simple vector
𝑶𝑶𝑶𝑶 in the common Cartesian
coordinates making an angle 𝜃𝜃
with the 𝑥𝑥-axis.
Here angular measure is used.
𝑂𝑂
𝐴𝐴
𝑥𝑥
𝜃𝜃
𝜋𝜋
2
=90°
𝑃𝑃
𝑦𝑦
𝑀𝑀
For the time
being, this is true
angular measure,
not radian
measure.

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Vector Components
This vector will generates two
projections, one on the 𝑥𝑥-axis as
OP and the other on the 𝑦𝑦-axis
as OM.
Projectors
𝑂𝑂
A
𝑥𝑥
𝜃𝜃 90°
𝑃𝑃
𝑦𝑦
Projectionon𝑦𝑦-axis
Projection on 𝑥𝑥-axis
𝑀𝑀

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Cos 𝜽𝜽 & sin 𝜽𝜽
These projection have a relation
with the initial position 𝑶𝑶𝑶𝑶
represented by the ratio of its
projection on the original
direction to the original length:
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅1 =
|𝑶𝑶𝑶𝑶|
|𝑶𝑶𝑶𝑶|
& 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅2 =
|𝑨𝑨𝑷𝑷|
|𝑶𝑶𝑶𝑶|
In trigonometry these ratios are
called the cosine (cos) and sine
(sin) of the angle 𝜃𝜃:
cos 𝜃𝜃 =
𝑂𝑂𝑂𝑂
𝑂𝑂𝑂𝑂
& 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 =
𝐴𝐴𝑃𝑃
𝑂𝑂𝑂𝑂
O
A
x
θ
P
y
Projectionon𝑦𝑦-axis
Projection on 𝑥𝑥-axis
M
cos 𝜃𝜃 =
𝑂𝑂𝑂𝑂
𝑂𝑂𝑂𝑂
𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 =
𝐴𝐴𝐴𝐴
𝑂𝑂𝑂𝑂

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A Normal Vector
In the normal Cartesian
coordinate systems, the vector
OA of length 𝑟𝑟 is written in
terms of its cosine and sine
components as:
𝑶𝑶𝑶𝑶 = 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 �𝒙𝒙 + 𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 �𝒚𝒚
Where �𝒙𝒙 and �𝒚𝒚 are the unit
vectors on the 𝑥𝑥 and 𝑦𝑦 axes
respectively.
𝑂𝑂
𝐴𝐴
𝑥𝑥
𝜃𝜃
𝑃𝑃
𝑦𝑦
𝑶𝑶𝑷𝑷 = 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
𝑀𝑀
𝑶𝑶𝑨𝑨 = 𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃

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POWER SERIES
Some functions of the variable 𝜃𝜃
such as cos 𝑥𝑥, ln 𝑥𝑥, √𝑥𝑥 etc., are
commonly occurring functions
and their values are found in the
familiar mathematical tables.
However unlike the other
simpler rational functions of 𝑥𝑥,
these values cannot be arrived
at by simple arithmetical
operations such as addition,
subtraction, multiplication, and
division. To obtain these values
as we see them in the tables, we
have to stretch them by using
power series.

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Maclaurin's series
In mathematics, a power series
is a sequence of variables in
different values of power,
generally in a properly arrange
order.
One of the power series of
relevant interest in our study is
the Maclaurin's series.
- - - - - - - -
Colin Maclaurin (1698 –1746) a
Scottish mathematician who
made important contributions to
geometry and algebra.[The
Maclaurin series, a special case
of the Taylor series, is named
after him.

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MACLAURIN'S SERIES
The Maclaurin's series is written as:
ƒ 𝑥𝑥 stands for the function of 𝑥𝑥.
The superscripts (′), ("), etc. stand for
the respective derivatives.
“!” stands for: 2! = 2 × 1; 5! = 5 ×
4 × 3 × 2 × 1.
ƒ(𝑥𝑥) = ƒ(0) + ƒ′(0)/1! 𝑥𝑥 + ƒ"(0)/2! 𝑥𝑥2
+ . . . . + 𝑓𝑓 𝑛𝑛
(0)/n! 𝑥𝑥 𝑛𝑛
+ . . .

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Maclaurin Series of sine and cosine
With the methodology of Maclaurin’s Series, the trigonometric
functions cos 𝑥𝑥 and sin 𝑥𝑥 can be expanded into the following
infinite series:
𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 = 1 −
𝑥𝑥2
2!
+
𝑥𝑥4
4!
−
𝑥𝑥6
6!
+ · · ·
𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 = 𝑥𝑥 −
𝑥𝑥3
3!
+
𝑥𝑥5
5!
−
𝑥𝑥7
7!
+ · · ·
3! = 3 × 2 × 1 5! = 5 × 4 × 3 × 2 × 1.

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Expansion of 𝑒𝑒 𝑥𝑥
Now we come back to our exponential friend 𝑒𝑒 𝑥𝑥
. It happened
that for 𝑒𝑒 raised to a variable power 𝑥𝑥 into 𝑒𝑒 𝑥𝑥
, there is also
an infinite series expression:
𝑒𝑒 𝑥𝑥
= 1 + 𝑥𝑥 +
𝑥𝑥2
2!
+
𝑥𝑥3
3!
+
𝑥𝑥4
4!
+
𝑥𝑥5
5!
+ · · ·

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Similarity
This is quite intriguing.
The functions of cosine and sine
are equations from the
trigonometry of right triangles.
On the other hand 𝑒𝑒 is an
exponentials that is related to
logarithms.
They are usually considered
totally separate and unrelated
areas in mathematics.
How can they be so similar in
these expressions?
Trigonometric:
𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 = 1 −
𝑥𝑥2
2!
+
𝑥𝑥4
4!
−
𝑥𝑥6
6!
+ · · ·
𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 = 𝑥𝑥 −
𝑥𝑥3
3!
+
𝑥𝑥5
5!
−
𝑥𝑥7
7!
+ · · ·
Exponential:
𝑒𝑒 𝑥𝑥
= 1 + 𝑥𝑥 +
𝑥𝑥2
2!
+
𝑥𝑥3
3!
+
𝑥𝑥4
4!
+
𝑥𝑥5
5!
+ · · ·

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More Striking Similarity
If we add the first two formulas, we get:
𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 + 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 = 1 + 𝑥𝑥 −
𝑥𝑥2
2!
−
𝑥𝑥3
3!
+
𝑥𝑥4
4!
+
𝑥𝑥5
5!
−
𝑥𝑥6
6!
−
𝑥𝑥7
7!
+ · · ·
Compare with the expansion of 𝑒𝑒:
𝑒𝑒 𝑥𝑥
= 1 + 𝑥𝑥 +
𝑥𝑥2
2!
+
𝑥𝑥3
3!
+
𝑥𝑥4
4!
+
𝑥𝑥5
5!
+
𝑥𝑥6
6!
+
𝑥𝑥7
7!
+ · · ·
The similarity are more pronounced.

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IN CAME EULER
To be continued on PM [B010]:
ABCC