PM [B09] The Trigo Companions

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THE TRIGO COMPANIONSPM [B09]


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 𝜃𝜃with the 𝑥𝑥-axis.Here angular measure is used.

𝑂𝑂

𝐴𝐴

𝑥𝑥

𝜃𝜃𝜋𝜋2

=90°

𝑃𝑃

𝑦𝑦

𝑀𝑀

For the time being, this is true angular measure, not radian measure.


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

Projection on𝑦𝑦-axis

Projection on 𝑥𝑥-axis

𝑀𝑀


Cos 𝜽𝜽 & sin 𝜽𝜽

These projection have a relation with the initial position 𝑶𝑶𝑶𝑶represented by the ratio of its projection on the original direction to the original length:

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅1 = |𝑶𝑶𝑶𝑶||𝑶𝑶𝑶𝑶|

& 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅2 = |𝑶𝑶𝑶𝑶||𝑶𝑶𝑶𝑶|

In trigonometry these ratios are called the cosine (cos) and sine(sin) of the angle 𝜃𝜃:

cos𝜃𝜃 = 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂

& 𝑠𝑠𝑅𝑅𝑠𝑠 𝜃𝜃 = 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂

O

A

x

θ

P

y

Projection on𝑦𝑦

-axis

Projection on 𝑥𝑥-axis

M

cos𝜃𝜃 =𝑂𝑂𝑃𝑃𝑂𝑂𝐴𝐴

𝑠𝑠𝑅𝑅𝑠𝑠 𝜃𝜃 =𝐴𝐴𝑃𝑃𝑂𝑂𝐴𝐴


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 𝑐𝑐𝑅𝑅𝑠𝑠 𝜃𝜃 �𝒙𝒙 + 𝑟𝑟 𝑠𝑠𝑅𝑅𝑠𝑠 𝜃𝜃 �𝒚𝒚

Where �𝒙𝒙 and �𝒚𝒚 are the unit vectors on the 𝑥𝑥 and 𝑦𝑦 axes respectively.

𝑂𝑂

𝐴𝐴

𝑥𝑥

𝜃𝜃

𝑃𝑃

𝑦𝑦

𝑶𝑶𝑶𝑶 = 𝑟𝑟 𝑐𝑐𝑅𝑅𝑠𝑠 𝜃𝜃

𝑀𝑀

𝑶𝑶𝑶𝑶 = 𝑟𝑟 𝑠𝑠𝑅𝑅𝑠𝑠 𝜃𝜃


POWER SERIES

Some functions of the variable 𝜃𝜃such 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.


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.


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 + . . . . + 𝑓𝑓𝑛𝑛(0)/n! 𝑥𝑥𝑛𝑛+ . . .


Maclaurin Series of sine and cosine

With the methodology of Maclaurin’s Series, the trigonometric functions cos 𝑥𝑥 and sin𝑥𝑥 can be expanded into the following infinite series:

𝑐𝑐𝑅𝑅𝑠𝑠 𝑥𝑥 = 1 −𝑥𝑥2

2!+𝑥𝑥4

4!−𝑥𝑥6

6!+ · · ·

𝑠𝑠𝑅𝑅𝑠𝑠 𝑥𝑥 = 𝑥𝑥 −𝑥𝑥3

3!+𝑥𝑥5

5!−𝑥𝑥7

7!+ · · ·

3! = 3 × 2 × 1 5! = 5 × 4 × 3 × 2 × 1.


Expansion of 𝑒𝑒𝑥𝑥

Now we come back to our exponential friend 𝑒𝑒𝑥𝑥. It happened that for 𝑒𝑒 raised to a variable power 𝑥𝑥 into 𝑒𝑒𝑥𝑥, there is also an infinite series expression:

𝑒𝑒𝑥𝑥 = 1 + 𝑥𝑥 +𝑥𝑥2

2!+𝑥𝑥3

3!+𝑥𝑥4

4!+𝑥𝑥5

5!+ · · ·


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:

𝑐𝑐𝑅𝑅𝑠𝑠 𝑥𝑥 = 1 −𝑥𝑥2

2!+𝑥𝑥4

4!−𝑥𝑥6

6!+ · · ·

𝑠𝑠𝑅𝑅𝑠𝑠 𝑥𝑥 = 𝑥𝑥 −𝑥𝑥3

3!+𝑥𝑥5

5!−𝑥𝑥7

7!+ · · ·

Exponential:


2!+𝑥𝑥3

3!+𝑥𝑥4

4!+𝑥𝑥5

5!+ · · ·


More Striking Similarity

If we add the first two formulas, we get:

𝑐𝑐𝑅𝑅𝑠𝑠 𝑥𝑥 + 𝑠𝑠𝑅𝑅𝑠𝑠 𝑥𝑥 = 1 + 𝑥𝑥 −𝑥𝑥2

2! −𝑥𝑥3

3! +𝑥𝑥4

4! +𝑥𝑥5

5! −𝑥𝑥6

6! −𝑥𝑥7

7! + · · ·

Compare with the expansion of 𝑒𝑒:


2! +𝑥𝑥3

3! +𝑥𝑥4

4! +𝑥𝑥5

5! +𝑥𝑥6

6! +𝑥𝑥7

7! + · · ·

The similarity are more pronounced.


IN CAME EULERTo be continued on PM [B010]:

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PM [B09] The Trigo Companions