Trigonometry. Hidden symmetry in base trigonometric functions. Part 2.

Definitions. It’s all about definitions. I prefer to discover a definition.

Lets go!

  • Assume that center of symmetry is angle of 90 degrees.
  • Convert angle from degree to radian and calculate:
    • sin(90)
    • cos(90)
    • tan(90)
  • If 90 is build of two numbers: nine and zero take two numbers from:
    • sin() after decimal point
    • cos()after decimal point
    • tan() – before decimal point and maybe after decimal point – i.e. when degree will be build of higher amount of numbers than amount numbers before decimal point
  • If number will be build of three numbers i.e. 911 you take three numbers
  • Sum all taken numbers.
  • The symmetry means the sum of numbers will be perfomed on degrees: 89 and 91 and that sums will be equal. The length of symmetry will be equal 1 – one number before and after 90 degrees.
  • When the sums will be equal at numbers 88 and 92 then length of symmetry will be equal 2 – two numbers before and after 90 degrees.

We will use some C++ code to show it.

Definitions:

const double PI = std::acos(-1);

double degreeToRadian(double degree) {
    return degree * PI / 180.0;
}

double radianToDegree(double radians) {
    return radians * 180 / PI;
}

double _sin(double x) {
    return std::sin(degreeToRadian(x));
}

double _cos(double x) {
    return std::cos(degreeToRadian(x));
}

double _tan(double x) {
    return std::tan(degreeToRadian(x));
}

Now we can calculate 89 and 91 degress:

int main() {
    std::cout.precision(20);

    std::cout << "sin(89) = " << _sin(89) << std::endl;
    std::cout << "cos(89) = " << _cos(89) << std::endl;
    std::cout << "tan(89) = " << _tan(89) << std::endl;

    std::cout << "sin(91) = " << _sin(91) << std::endl;
    std::cout << "cos(91) = " << _cos(91) << std::endl;
    std::cout << "tan(91) = " << _tan(91) << std::endl;

    return 0;
}

Result will be:

sin(89) = 0.99984769515639126958
cos(89) = 0.017452406437283376345
tan(89) = 57.289961630759876243
sin(91) = 0.99984769515639126958
cos(91) = -0.017452406437283476959
tan(91) = -57.289961630759549394

So what now? We can see:

sin(89) = sin(91)
abs(cos(89)) = abs(cos(91))
abs(tan(89)) = abs(tan(91))

And as we define above we take numbers from trigonometry functions from degree 89 and degree 91.
9 + 9 + 0 + 1 + 5 + 7 = 31
9 + 9 + 0 + 1 + 5 + 7 = 31
Symmetry exists because two sums are equal.
When the symmetry is calculated between multipliers of 90 degrees then symmetry should exists.

But what if I tell you the secret if the symmetry exists when the center is available on 102 degress?

Let us check it.

int main() {
    std::cout.precision(20);

    std::cout << "sin(101) = " << _sin(101) << std::endl;
    std::cout << "cos(101) = " << _cos(101) << std::endl;
    std::cout << "tan(101) = " << _tan(101) << std::endl;

    std::cout << "sin(103) = " << _sin(103) << std::endl;
    std::cout << "cos(103) = " << _cos(103) << std::endl;
    std::cout << "tan(103) = " << _tan(103) << std::endl;

    return 0;
}

You will get:

sin(101) = 0.98162718344766397571
cos(101) = -0.19080899537654480436
tan(101) = -5.1445540159703107008
sin(103) = 0.97437006478523524589
cos(103) = -0.22495105434386480914
tan(103) = -4.3314758742841590333

You will find:

sin(101) != sin(103)
cos(101) != cos(103)
tan(101) != tan(101)

So where is that symmetry?

As we are following the previous definitions:

  • from sin(101) take: “9”, “8”, “1”: after decimal separator
  • from cos(101) take: “1”, “9”, “0”: after decimal separator
  • from tan(101) take “5”, “1”, “4”: before decimal separator and after

and

  • from sin(103) take: “9”, “7”, “4”: after decimal separator
    from cos(103) take: “2”, “2”, “4”: after decimal separator
    from tan(103) take “4”, “3”, “3”: before decimal separator and after

From _numbers(101) you will build a sum:
9 + 8 + 1 +
1 + 9 + 0 +
5 + 1 + 4 = 38
And from _numbers(103) you will build a sum:
9 + 7 + 4 +
2 + 2 + 4 +
4 + 3 + 3 = 38

So it means:

sum(_numbers(101)) = sum(_numbers(103))

So it is fact:

When angle number is build of 4 numbers (i.e. “1337”) or any other quantity of numbers – using the same amount of numbers to calculate symmetry from sum(_numbers(x)) will create real symmetry results!

Also we can observe that symmetries are not only at multipliers of 90 degress but at other degree values with different symmetry lengths.

Pretty clever, huh?

I am open for suggestions how it can be possible.

At incoming Part 3. I will introduce c++ code and result text files for larger range of numbers.

Code for this part can be found at Github.

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