The calculator

[Mikl__]

Ola, RuiLoureiro!
ORN = (Not A) OR (Not B) = Not (A AND B) ?

[dedndave]

Code Select Expand

NAND = NOT (A AND B)
NOR = NOT (A OR B)
ORN = A OR (NOT B)

[RuiLoureiro]

Olá Mikl,
          It's not right.
          I use A ORN B = A or (not B) = A or -B as Dave wrote
         
Hi
        In the next Calcula55, The powerfull Calculator v2.30,

        we may solve any quadratic or any system of 2,3,4 equations

        where any coeficient may be a real expression as large as

        a train (11 300 characters)!

        Well, we may solve any system of 5,6, ..,19,20 equations

        using matrices;

        We may define or redefine any one of 20 constants: real, logic

        or matrix.

        Each matrix element is a real constant or any real expression

        as large as a train!
         
        When we want to compute a real/logic expression, we may define or

        redefine a constant:

                        (log(2)+56.38)/12.5        (=4.5344823996531185)
                or
                        this = (log(2)+56.38)/12.5 (=4.5344823996531185)

       (the expression may contain real constants)

        Using 'list r' we may edit the real constants and we may redefine them.

        Using 'list l' we may edit the logic constants and we may redefine them.

        Any 20x20 matrix fits in th edit box, so we may copy it from or to

        the edit box.

        What do you think about this version ?

        Thanks
         Rui
-------------------------------------------------------------------------
TYPE:   this = (log(2)+56.38)/12.5      ENTER/COMPUTE

TYPE:   a=12.3;b=-1.5;c=15;d=3.24;      ENTER/COMPUTE


Quadratic equation:

TYPE:   ax^2+bx+c=d                     ENTER/COMPUTE

Root X0=  0.0609756097560975+ i  0.9758993472640925
Root X1=  0.0609756097560975- i  0.9758993472640925

--------------------------------------------------------------------------
Quadratic equation:

TYPE:   this x^2 + bx + c=d             ENTER/COMPUTE

Root X0=  0.1653992526373845+ i  1.6019061672211593
Root X1=  0.1653992526373845- i  1.6019061672211593
--------------------------------------------------------------------------
System of 2 equations:

TYPE:   this x + by= c ; x - d y= 12;

X=  2.3196363400200491
Y= -2.9877665617222071
Determinant: -13.1917229748761039

[RuiLoureiro]

    ****-- Here it is The Calculator v3.00.1 --****

       This is the new calculator with new functions
       and new better and faster procedures.

       A. What it does ?
       ---------------------

        1. Conversions

        2. Real constants definitions

        3. Logic constants definitions     

        4. Function definition

        5. Derivative definition

        6. Matrices definitions

        7. Logic operations

        8. Matrices operations

        9. Systems of 2 linear equations x,y
                       
       10. Systems of 3 linear equations x,y,z

       11. Systems of 4 linear equations x,y,z,t

       12. Quadratic equation ax^2+bx+c=d

       13. Solves any real expression


       B. Special functions
       ------------------------

        B.1. list     shows the symbols defined

             list r   edit the r eal constans defined

             list l   edit the l ogic constants defined

             list f   edit the f unction defined

             list d   edit the d erivative defined


        B.2. scan     shows the values of a defined function f(x)     
                      from x=x0 to x=x1 where x0,x1 are integers.
                      It shows also the zeros

        B.3. root     try to find the value x=x0 where the function f(x)=0


       C. General rule about constants and matices
       --------------------------------------------------------

            1. Any constant or matrix is redefinable
           
            2. The tables have space for 20 names


       C. General rule about real constants
       ----------------------------------------------

            1. The name cannot be any symbol already defined
               like 'e' 'pi', 'log', 'sin', etc.

            2. The name cannot be x, y, z, t

            3. The name cannot be a matrix name

            4. The matrix name cannot be any symbol already defined
               and cannot be any real constant name

            5. Any real constant is redefinable

            6. We may define a real constant like this

                            name = real expression


       D. General rule about logic constants
       ----------------------------------------------

            1. The name cannot be any symbol already defined

            2. The name cannot be 'd', 'h', 'b'

            3. We may use real constant names or matrix names
               because logic expressions doesnt use real values
               or matrix names and real expressions doesnt use
               logic constants
       

       E. General rule about expressions
       -------------------------------------------

            We may use any constant already defined
                       

       F. Defining a matrix
       -------------------------

            F.1
                    matA =[1,2,3; 3,4,5; 2,5,9]         (without semicolon)


            F.2
                    a=2; b=3;
                    matB =[3*2, a, b; 1.2, 4.5,-5.23];  (with semicolon)


            F.3
                    matC =[2, a, b; 1, 5,-5; 0,1,-1];
                    matC =matC^-1;
           

       G. Defining a real constant
       ----------------------------------

            G.1
                    a=2;b=-5;c=3;

            G.2
                    a=2;b=3;
                    f=round(e^-(-3+sin(3*a+12)),6)+34*5-sin(pi/b)


       H. Defining a logic constant
       ----------------------------------
     

            H.1
                    l1=2d; l2=53h; l3=11100011b;

            H.2
                    l1=2d; l2=53h; l3=11100011b;

                    l4= l1 + l2 shl 1d + 45h and l3                   


       I. System of equations
       -----------------------------

            1.      a=2;b=-3;c=4;d=-1;f=5;g=-2;h=-6;i=0;j=4;k=1;l=-3;

            2.      aX-bY=c; dY+fX=-a;
                   
                    aX-bY+cZ=d; Z-dY+fX=-a; X+Y+Z=h;
                   
                    kT+aX-bY+cZ=d; Z-dY+fX=-a; X+Y+Z-T=h; Y+lZ+fT=l;

       J. Quadratic equation
       ---------------------------

                    a=2;b=-3;c=4;d=-1;
                    aX^2 -bX +c=d                   

       K. root function
       --------------------

            1. f(x)=2*x^5+3*x^4-x^3+5*x^2+x+120
                           

            2. root(x=-2,x=2)

               root(x=-2,x=2; n=1000)

               root(x=-2,x=2; d=0.01)


               df(x)=10*x^4+12*x^3-3*x^2+10*x+1
               root(x=-2,x=2; x=-2)


       L. scan function
       --------------------

            1. f(x)=x^5-5*x^3+4*x
                           

            2. scan(x=-10,x=10)


            Now type 'list' and we can see all constants defined till now.


       M. conversions
       -------------------

                Type 112233d or 112233h or 111000111b and press COMPUTE
               
        note 1: type/copy and paste the examples and press compute
        note 2: to get the polynomials zeros use scan() function

        Try it and say something.
        Good luck !
        Thanks       
        Rui Loureiro

[RuiLoureiro]

    ****-- The Calculator v3.00.2 --****
    ****--     The new version    --****

    --------------------------------   
    Three things  A.1  A.2  A.3
    --------------------------------

            --------------------------------------------------------------------
       A.1. Define a matrix as a direct operation of 2 matrices
            ---------------------------------------------------------------------

            matA= -1.2*[1,2,3; 4,5,6; 7,8,9]

            matB= [1,2,3; 4,5,6; 7,8,9]*[3; 6; 9]

            matC= [1,2,3; 4,5,6; 7,8,9]-[-9,8,7; -4,0,6; 1,3,2]

    Example 1
   
        Solving a system of 5 linear equations:

Code Select Expand


            1 X1 -2 X2 +3 X3 -1 X4 + 0 X5 = 123

            0 X1 +3 X2 -2 X3 +5 X4 + 7 X5 = 15

            3 X1 -4 X2 +0 X3 -2 X4 + 3 X5 = 9

           -2 X1 -2 X2 +5 X3 +0 X4 + 0 X5 = -12

            1 X1 +0 X2 -1 X3 +3 X4 + 9 X5 = 35


        matC=[123; 15; 9; -12; 35]
        matA=[1,-2,3,-1,0; 0,3,-2,5,7; 3,-4,0,-2,3; -2,-2,5,0,0; 1,0,-1,3,9]

        matA1=matA^-1;
             =[ 0.5230125523012552, 0.2761506276150627, 0.0711297071129707,
             -0.2510460251046025,-0.2384937238493723; 0.6903765690376569,
             -0.7154811715481171,-0.8661087866108786,-0.5313807531380753,
             0.8451882845188284; 0.4853556485355648,-0.1757322175732217,
             -0.3179916317991631,-0.11297071129707113, 0.2426778242677824;
             -0.401673640167364, 1.1799163179916318, 0.8493723849372384,
             0.4728033472803347,-1.200836820083682; 0.1297071129707113,
             -0.4435146443514644,-0.3263598326359832,-0.1422594142259414,
             0.5648535564853556];

        multiplying by [123; 15; 9; -12; 35]
               
            = [ 63.778242677824266; 102.347280334728032; 64.050209205020915;
              -71.765690376569036; 27.841004184100418];

            X1=  63.778242677824266
            X2= 102.347280334728032
            X3=  64.050209205020915
            X4= -71.765690376569036
            X5=  27.841004184100418


        first equation:
       
            63.778242677824266 -2* 102.347280334728032 + 3* 64.050209205020915
              +71.765690376569036 = 122.999999999999983


             --------------------
        A.2 find function
             --------------------

             Now we have the scan function and the find function
             to study one function.
             
             The scan function uses integer points and is useful to
             study any polynomial.
             
             The find function try to get an interval (x0,x1) where
             the function changes the sign. So it may be a discontinuity
             or it may be a zero (it shows the f(x) values)

    Example 2

Code Select Expand


f(x)=(x+1)/(x^2-x-1)

find(x=-10, x=10)
One zero was found
x= -1.0
f(x)=  0


    Example 3

Code Select Expand


f(x)=(log(x-1)-x)/(x-5)

find(x=1, x=20)
The function change the sign in this interval
(x0,x1)=[  4.9999 ,  5.0 ]
f(x0)=  43978.50866169812
f(x1)= -INFINITY


    Example 4

Code Select Expand


f(x)=x-log(1/(x-1))

find(x=1, x=5)
The function change the sign in this interval
(x0,x1)=[  1.0826 ,  1.0827 ]
f(x0)= -0.0004199526796177
f(x1)=  0.0002055095525466


    Example 5

Code Select Expand


f(x)=(x-log(1/(x-1)))/(x^2-1)

find(x=1, x=10)
The function change the sign in this interval
(x0,x1)=[  1.0826 ,  1.0827 ]
f(x0)= -0.0004199526796177
f(x1)=  0.0002055095525466



    Example 6

Code Select Expand


f(x)=x^3-log(x^2+x)+x-1

find(x=0,x=1)
The function change the sign in this interval
(x0,x1)=[  0.1177 ,  0.1178 ]
f(x0)=  0.0002288184583431
f(x1)= -0.000074703978969

scan(x=-2,x=2)
x= -2 , f(x)=-11.3010299956639812 ; x= -1 , f(x)=+INFINITY ;
x= 0 , f(x)=+INFINITY ; x= 1 , f(x)= 0.6989700043360188 ;
x= 2 , f(x)= 8.2218487496163564 ;



             ---------------------------------------------------------------------------
        A.3  The limits of the root/find function may be real expressions
             ---------------------------------------------------------------------------

             f(x)=x^3-x

             root(x=log(2), x=log(3))
             find(x=-log(3), x=log(3))

        Good luck !
        Rui Loureiro

[RuiLoureiro]

    ****-- The Calculator v3.01.0 --****

            ---------------------------
            COMPLEX NUMBERS
            ---------------------------

        Now, we may solve any complex expression in the same way we do for
        real expressions.

        Any complex number is a number with a real part
        and an imaginary part. For instance, z1=-2+i3 or z1=i3-2.

        We may define it typing z1=(-2,3); or z1=-2+i3 for example.

        To build a complex expression
        we may define a set of real/complex constants z1, z2, z3, z4, ...
        and then we write the expression

        For instance,

                (1-i2)*(2-i3)  or (a-ib)*(c-id) where a=12;b=-4;c=1;d=-3;
                                                are real constants

                z1+z2*z3-z4   or  z5=z1+z2*z3-z4  or z1=z1+z2*(z3/z4)+3-i2


        Type:   d=7;c=1;b=-3;a=2;    press ENTER/COMPUTE
        and     (a-ib)*(c-id)                 press ENTER/COMPUTE
        and     Z1=(a-ib)*(c-id)           press ENTER/COMPUTE

        We get:      23.0-i11.0
                     | Z |=  25.495097567963924
                     Angle= -25.559965171823808 degrees
                     Complex Constant defined


        If you have any suggestion, please tell me

        Jul. 2013
       
        Good luck !
        Thanks       
        Rui Loureiro
                            EDIT: In the next version the calculator will do all
                                      complex functions (e.g.: ln(2-i3), etc.  )

[RuiLoureiro]

Hi all
        ****-- The Calculator v3.10.1 --****

        This is the last version and this is the powerful calculator
        v3.10.1.

            --------------------------------------------------------------
            COMPLEX NUMBERS and COMPLEX FUNCTIONS
            --------------------------------------------------------------

        Now, we may solve any complex expression in the same way we do for
        real expressions.

        We may use any function:

                conj(a+ib)      = a-ib
                inv(a+ib)       = 1/(a+ib)

                abs(a+ib)       = sqr(a^2+b^2)+i0
                sqr(a+ib)       = (a+ib)^(1/2)
                rnd(a+ib)       = rnd(a)+i rnd(b)
                rndi(a+ib)      = rndi(a)+i rndi(b)
       
                e^(a+ib)        = e^a * e^ib = e^a * ( cos(b)+i sin(b))

                (a+ib)^(c+id)   = e^( (c+id)*ln(a+ib) )
               
                ln(a+ib)        = ln (|a+ib|* e^i angleR )
                log(a+ib)       = (ln (|a+ib|* e^i angleR )) / ln(10)
               
                sin(a+ib)
                cos(a+ib)
                tan(a+ib)
                sec(a+ib)
                csc(a+ib)
                cot(a+ib)

                arcsin(a+ib)
                arccos(a+ib)
                arctan(a+ib)
                arcsec(a+ib)
                arccsc(a+ib)
                arccot(a+ib)

                sinh(a+ib)
                cosh(a+ib)
                tanh(a+ib)
                sech(a+ib)
                csch(a+ib)
                coth(a+ib)

                arcsinh(a+ib)
                arccosh(a+ib)
                arctanh(a+ib)
                arcsech(a+ib)
                arccsch(a+ib)
                arccoth(a+ib)
               
                round(a+ib, n)  - round 'a' and 'b' to n decimal places

        The operation rules are the same for real numbers

        The result may be a complex number, INFINITY or indeterminate form.
        We may use the division by 0 to generate the infinity.

        We need to use brackets when we have powers of powers

        (1-i2)^(1-i)^(i2) gives "Complex power too complex- use brackets"

        We need to do ((1-i2)^(1-i))^(i2) or (1-i2)^((1-i)^(i2)).

        Any complex number is a number with a real part 'a'
        and an imaginary part 'b'. For instance, z1=-2+i3 or z1=i3-2.

        We may define it typing z1=(-2,3); or z1=-2+i3 for example.

        To build a complex expression
        we may define a set of real/complex constants z1, z2, z3, z4, ...
        and then we write the expression

        For instance,

                (1-i2)*(2-i3)  or (a-ib)*(c-id) where a=12;b=-4;c=1;d=-3;
                                                are real constants

                z1+z2*z3-z4   or  z5=z1+z2*z3-z4  or z1=z1+z2*(z3/z4)+3-i2

        After defining any constant/matrix we may type the constant name
        to see its value. We may use also list c or list r or list l or
        list.

        Example 1
       
        Type:   d=7;c=1;b=-3;a=2;    press ENTER/COMPUTE
        and     (a-ib)*(c-id)        press ENTER/COMPUTE
        and     Z1=(a-ib)*(c-id)     press ENTER/COMPUTE

        We get:      23.0-i11.0
                     | Z |=  25.495097567963924
                     Angle= -25.559965171823808 degrees
                     Angle= -0.4461055489434036 radians
                     Complex Constant defined

        Example 2

            conj(-1-i2)+inv(1-i3)

            -0.9+i2.3
            | Z |=  2.4698178070456938
            Angle=  111.370622269343183 degrees
            Angle=  1.9437840485949576 radians


        Example 3

            zx=arctan(-1-i3.3)+sin(1.2+i3e-2)*(i*cos(1-i))^(2-i)
           
            -1.2156119789977935-i18.15086513723519
            | Z |=  18.191525986392202
            Angle= -93.831529845398134 degrees
            Angle= -1.6376691379855234 radians
            Complex Constant defined

        Example 4

            zy=(ln(zx)+1-i)* (ln(zx-1)+1+i)+e^(9.1-i2.3e-1)  ; edit: ')' was not here
           
            8732.8791809611277-i2054.6089708474515
            | Z |=  8971.3207953094131
            Angle= -13.2393627831433264 degrees
            Angle= -0.231070471431851 radians
            Complex Constant defined

        Example 5

            zz=zx*zy-zy/zx
            -47989.335119512577-i156498.23124802209
            | Z |=  163690.78370199185
            Angle= -107.0478745161774 degrees
            Angle= -1.8683378675690276 radians
            Complex Constant defined

        Example 6

            zw=(e+i pi)*(pi-ie)/((ipi)* ie)

            -2.0-i0.2904713703586566
            | Z |=  2.0209833292231868
            Angle= -171.73638751817216 degrees
            Angle= -2.9973654076729973 radians
            Complex Constant defined

        Example 7

            zr=round( (e+i pi)*(pi-ie)/((ipi)* ie), 3)
                       
            -2.0-i0.29
            | Z |=  2.0209156340629363
            Angle= -171.7496127710945 degrees
            Angle= -2.9975962318809012 radians
            Complex Constant defined

        If you know any bug or something else, please post it
       
        If you have any suggestion, please tell me

        Jul. 2013
       
        Good luck !  ;)
        Thanks       
        Rui Loureiro

[guga]

Trabalho excelente, rui.

Me ajudou muito há pouco quando estava testando uma função de atan2.

[RuiLoureiro]

Olá guga,
          obrigado !
Hi all,

        (1) The previous The Calculator v3.10.1 (calcula59.exe)
            dosn't solve trigonometric functions,
            when we are working with angles in DEGREES:

                sind(a+ib)
                cosd(a+ib)
                tand(a+ib)
                secd(a+ib)
                cscd(a+ib)
                cotd(a+ib)

                arcsind(a+ib)
                arccosd(a+ib)
                arctand(a+ib)
                arcsecd(a+ib)
                arccscd(a+ib)
                arccotd(a+ib)

           If we try, we get "Complex expression error"

        (2) The results of some complex trigonometric functions
            are not correct
            because there is a bug in one internal procedure
            (i wanted «fstp OperandZ» but i did «fstp OperandI» !!!)

        (3) there is a problem when we try to define variables x1, x2, x3, etc

            All these problems are solved in
            the NEXT version v3.10.2 (calcula60.exe)
            that i will post soon.

--------------------------------------------------------------------
These are some values that we get, when we use the
next version v3.10.2 (calcula60.exe)
--------------------------------------------------------------------
--------------------------------------
about trigonometric functions
inverse trigonometric functions
RADIANS
--------------------------------------

Code Select Expand


TYPE:   z1=pi/4+i3
WE GET:

0.7853981633974483+i3.0
| Z |=  3.101104686247803
Angle=  75.329256947468645 degrees
Angle=  1.314743556814141 radians
Complex Constant defined

TYPE:   z2=sin(z1)
WE GET:
7.1189120679085489+i7.0837072942502342
| Z |=  10.0427993139974647
Angle=  44.857978252688267 degrees
Angle=  0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arcsin(z2)
WE GET:
0.7853981633974483+i3.0                 ->>>> is = z1 (correct)
| Z |=  3.101104686247803
Angle=  75.329256947468645 degrees
Angle=  1.314743556814141 radians
Complex Constant defined

TYPE:   z2=cos(z1)
WE GET:
7.1189120679085491-i7.0837072942502341
| Z |=  10.0427993139974647
Angle= -44.857978252688266 degrees
Angle= -0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arccos(z2)
WE GET:
0.7853981633974483+i3.0                 ->>>> is = z1 (correct)
| Z |=  3.101104686247803
Angle=  75.329256947468645 degrees
Angle=  1.314743556814141 radians
Complex Constant defined

TYPE:   z2=tan(z1)
WE GET:
0.0049574738935603+i0.9999877116507955
| Z |=  1.0
Angle=  89.715956505376533 degrees
Angle=  1.5658388325948463 radians
Complex Constant defined

TYPE:   w1=arctan(z2)
WE GET:
0.7853981633974411+i3.0000000000000079  ->>>> is = z1 (correct)
| Z |=  3.1011046862478089
Angle=  75.32925694746881 degrees
Angle=  1.3147435568141439 radians
Complex Constant defined

TYPE:   z2=csc(z1)
WE GET:
0.0705836414544171-i0.0702345879617375
| Z |=  0.099573830834817
Angle= -44.857978252688267 degrees
Angle= -0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arccsc(z2)
WE GET:
0.7853981633974483+i3.0000000000000004  ->>>> is = z1 (correct)
| Z |=  3.1011046862478034
Angle=  75.329256947468646 degrees
Angle=  1.3147435568141411 radians
Complex Constant defined

TYPE:   z2=sec(z1)
WE GET:
0.0705836414544171+i0.0702345879617375
| Z |=  0.099573830834817
Angle=  44.857978252688266 degrees
Angle=  0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arcsec(z2)
WE GET:
0.7853981633974482+i3.0000000000000004  ->>>> is = z1 (correct)
| Z |=  3.1011046862478034
Angle=  75.329256947468647 degrees
Angle=  1.3147435568141411 radians
Complex Constant defined

TYPE:   z2=cot(z1)
WE GET:
0.0049574738935603-i0.9999877116507955
| Z |=  1.0
Angle= -89.715956505376533 degrees
Angle= -1.5658388325948463 radians
Complex Constant defined

TYPE:   w1=arccot(z2)
WE GET:
0.7853981633974554+i3.0000000000000079  ->>>> is = z1 (correct)
| Z |=  3.1011046862478125
Angle=  75.329256947468554 degrees
Angle=  1.3147435568141395 radians
Complex Constant defined


--------------------------------------
about trigonometric functions
inverse trigonometric functions
RADIANS
--------------------------------------

Code Select Expand


--------------
First quadrant
--------------
z1=pi/4+i3    = 0.7853981633974483+i3.0

y1=sin(z1)    = 7.1189120679085489+i7.0837072942502342
w1=arcsin(y1) = 0.7853981633974483+i3.0 = z1

y1=cos(z1)    = 7.1189120679085491-i7.0837072942502341
w1=arccos(y1) = 0.7853981633974483+i3.0 = z1

y1=tan(z1)    = 0.0049574738935603+i0.9999877116507955
w1=arctan(y1) = 0.78539816339744 11+i3.00000000000000 79 = z1

---------------
Second quadrant
---------------
z2=-pi/4+i3   = -0.7853981633974483+i3.0

y2=sin(z2)    = -7.1189120679085489+i7.0837072942502342
w2=arcsin(y2) = -0.7853981633974483+i3.0 = z2

y2=cos(z2)    = 7.1189120679085491+i7.0837072942502341
w2=arccos(y2) = -0.7853981633974483+i3.0 = z2

y2=tan(z2)    = -0.0049574738935603+i0.9999877116507955
w2=arctan(y2) = -0.78539816339744 11+i3.00000000000000 79 = z2

--------------
Third quadrant
--------------
z3=-pi/4-i3 = -z1

y3=sin(z3)    = -7.1189120679085489-i7.0837072942502342
w3=arcsin(y3) = -0.7853981633974483-i3.0 = z3

y3=cos(z3)    = 7.1189120679085491-i7.0837072942502341
w3=arccos(y3) =  0.7853981633974483+i3.0 = z1

y3=tan(z3)    = -0.0049574738935603-i0.9999877116507955
w3=arctan(y3) = -0.78539816339744 11-i3.00000000000000 79 = z3

---------------
Fourth quadrant
---------------
z4=pi/4-i3  = -z2

y4=sin(z4)    = 7.1189120679085489-i7.0837072942502342
w4=arcsin(y4) = 0.7853981633974483-i3.0 = z4

y4=cos(z4)    = 7.1189120679085491+i7.0837072942502341
w4=arccos(y4) = -0.7853981633974483+i3.0 = z2

y4=tan(z4)    = 0.0049574738935603-i0.9999877116507955
w4=arctan(y4) = 0.78539816339744 11-i3.00000000000000 79 = z4


--------------------------------------
about trigonometric functions
inverse trigonometric functions
DEGREES
--------------------------------------

Code Select Expand


TYPE:   z1=45+i3
WE GET:
| Z |=  45.099889135118725
Angle=  3.8140748342903542 degrees
Angle=  0.0665681637758238 radians
Complex Constant defined

TYPE:   z2=sind(z1)
WE GET:
7.118912067908549+i7.0837072942502342
| Z |=  10.0427993139974647
Angle=  44.857978252688266 degrees
Angle=  0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arcsind(z2)
WE GET:
45.0+i3.0                           ->>>> is = z1 (correct)
| Z |=  45.099889135118724
Angle=  3.8140748342903542 degrees
Angle=  0.0665681637758238 radians
Complex Constant defined

TYPE:   z2=cosd(z1)
WE GET:
7.118912067908549-i7.0837072942502342
| Z |=  10.0427993139974647
Angle= -44.857978252688266 degrees
Angle= -0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arccosd(z2)
WE GET:
45.0+i3.0                           ->>>> is = z1 (correct)
| Z |=  45.099889135118725
Angle=  3.8140748342903542 degrees
Angle=  0.0665681637758238 radians
Complex Constant defined

TYPE:   z2=tand(z1)
WE GET:
0.0049574738935603+i0.9999877116507955
| Z |=  1.0
Angle=  89.715956505376533 degrees
Angle=  1.5658388325948463 radians
Complex Constant defined

TYPE:   w1=arctand(z2)
WE GET:
44.99999999999959+i3.0000000000000079   ->>>> is = z1 (correct)
| Z |=  45.099889135118316
Angle=  3.8140748342903989 degrees
Angle=  0.0665681637758245 radians
Complex Constant defined

TYPE:   z2=cscd(z1)
WE GET:
0.0705836414544171-i0.0702345879617375
| Z |=  0.099573830834817
Angle= -44.857978252688266 degrees
Angle= -0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arccscd(z2)
WE GET:
45.000000000000002+i3.0000000000000004  ->>>> is = z1 (correct)
| Z |=  45.099889135118727
Angle=  3.8140748342903545 degrees
Angle=  0.0665681637758238 radians
Complex Constant defined

TYPE:   z2=secd(z1)
WE GET:
0.0705836414544171+i0.0702345879617375
| Z |=  0.099573830834817
Angle=  44.857978252688266 degrees
Angle=  0.7829194162974231 radians
Complex Constant defined

TYPE:   w1=arcsecd(z2)
WE GET:
44.999999999999998+i3.0000000000000004  ->>>> is = z1 (correct)
| Z |=  45.099889135118723
Angle=  3.8140748342903549 degrees
Angle=  0.0665681637758238 radians
Complex Constant defined

TYPE:   z2=cotd(z1)
WE GET:
0.0049574738935603-i0.9999877116507955
| Z |=  1.0
Angle= -89.715956505376533 degrees
Angle= -1.5658388325948463 radians
Complex Constant defined

TYPE:   w1=arccotd(z2)
WE GET:
45.00000000000041+i3.0000000000000079   ->>>> is = z1 (correct)
| Z |=  45.099889135119134
Angle=  3.8140748342903296 degrees
Angle=  0.0665681637758233 radians
Complex Constant defined


---------------------------------
about hyperbolic functions
inverse hyperbolic functions
---------------------------------

Code Select Expand


TYPE:   z1=2.3+i0.25
WE GET:
2.3+i0.25
| Z |=  2.3135470602518549
Angle=  6.2034479016918352 degrees
Angle=  0.10827059086045604 radians
Complex Constant defined

TYPE:   z2=sinh(z1)
WE GET:
4.783483618904492+i1.2462283322677304
| Z |=  4.9431569455636773
Angle=  14.6025145321957282 degrees
Angle=  0.2548619576571349 radians
Complex Constant defined

TYPE:   w1=arcsinh(z2)
WE GET:
2.3+i0.25                               ->>>> is = z1 (correct)
| Z |=  2.3135470602518549
Angle=  6.2034479016918353 degrees
Angle=  0.10827059086045604 radians
Complex Constant defined

TYPE:   z2=cosh(z1)
WE GET:
4.8806256579738629+i1.2214238973804283
| Z |=  5.0311413367510157
Angle=  14.0502468762936514 degrees
Angle=  0.2452230687093726 radians
Complex Constant defined

TYPE:   w1=arccosh(z2)
WE GET:
2.3+i0.25                               ->>>> is = z1 (correct)
| Z |=  2.3135470602518549
Angle=  6.2034479016918353 degrees
Angle=  0.10827059086045604 radians
Complex Constant defined

TYPE:   z2=tanh(z1)
WE GET:
0.9824664000943306+i0.0094701778128957
| Z |=  0.9825120414438294
Angle=  0.5522676559020767 degrees
Angle=  0.0096388889477623 radians
Complex Constant defined

TYPE:   w1=arctanh(z2)
WE GET:
2.3000000000000002+i0.249999999999998   ->>>> is = z1 (correct)
| Z |=  2.3135470602518549
Angle=  6.2034479016917859 degrees
Angle=  0.10827059086045518 radians
Complex Constant defined

TYPE:   z2=csch(z1)
WE GET:
0.1957651997848019-i0.0510021895924765
| Z |=  0.2022998684873777
Angle= -14.6025145321957282 degrees
Angle= -0.2548619576571349 radians
Complex Constant defined

TYPE:   w1=arccsch(z2)
WE GET:
2.3000000000000003+i0.2499999999999997  ->>>> is = z1 (correct)
| Z |=  2.3135470602518552
Angle=  6.2034479016918292 degrees
Angle=  0.10827059086045593 radians
Complex Constant defined

TYPE:   z2=sech(z1)
WE GET:
0.1928157309006095-i0.0482540063543098
| Z |=  0.1987620567713517
Angle= -14.0502468762936514 degrees
Angle= -0.2452230687093726 radians
Complex Constant defined

TYPE:   w1=arcsech(z2)
WE GET:
2.3000000000000005+i0.2499999999999998  ->>>> is = z1 (correct)
| Z |=  2.3135470602518554
Angle=  6.2034479016918313 degrees
Angle=  0.10827059086045597 radians
Complex Constant defined

TYPE:   z2=coth(z1)
WE GET:
1.017751950195062-i0.0098103018452776
| Z |=  1.0177992307661405
Angle= -0.5522676559020767 degrees
Angle= -0.0096388889477623 radians
Complex Constant defined

TYPE:   w1=arccoth(z2)
WE GET:
2.3000000000000003+i0.2499999999999993  ->>>> is = z1 (correct)
| Z |=  2.3135470602518551
Angle=  6.2034479016918193 degrees
Angle=  0.10827059086045576 radians
Complex Constant defined


---------------------------------
hyperbolic functions
inverse hyperbolic functions
---------------------------------

Code Select Expand


TYPE:   z1=2.3+i1.58 = Z + iI  ( I > pi/2)
WE GET:
2.3+i1.58
| Z |=  2.790412155936825
Angle=  34.487372830078654 degrees
Angle=  0.6019182062477074 radians
Complex Constant defined

TYPE:   y1=sinh(z1)
WE GET:
-0.045437541593796+i5.0370073053376468
| Z |=  5.0372122413306062
Angle=  90.516836405985953 degrees
Angle=  1.5798168237735256 radians
Complex Constant defined

TYPE:   w1=arcsinh(y1)
WE GET:
-2.3+i1.5615926535897932     -> is not = z1
| Z |=  2.7800308659699467
Angle=  145.82532569791053 degrees
Angle=  2.5451320662216368 radians
Complex Constant defined


---------------------------------
about hyperbolic functions
inverse hyperbolic functions
---------------------------------

Code Select Expand



--------------
First quadrant
--------------
z1=2.3+i0.25

y1=sinh(z1)    = 4.783483618904492+i1.2462283322677304
w1=arcsinh(y1) =  2.3+i0.25 = z1

y1=cosh(z1)    = 4.8806256579738629+i1.2214238973804283
w1=arccosh(y1) = 2.3+i0.25 = z1

y1=tanh(z1)    = 0.9824664000943306+i0.0094701778128957
w1=arctanh(y1) = 2.3000000000000002+i0.249999999999998 = z1

---------------
Second quadrant
---------------
z2=-2.3+i0.25  = -z4

y2=sinh(z2)    = -4.783483618904492+i1.2462283322677304
w2=arcsinh(y2) = -2.3+i0.25 = z2

y2=cosh(z2)    = 4.8806256579738629-i1.2214238973804283
w2=arccosh(y2) = 2.3-i0.25 = z4

y2=tanh(z2)    = -0.9824664000943306+i0.0094701778128957
w2=arctanh(y2) = -2.3000000000000002+i0.249999999999998 = z2

--------------
Third quadrant
--------------
z3=-2.3-i0.25 = -z1

y3=sinh(z3)    = -4.783483618904492-i1.2462283322677304
w3=arcsinh(y3) = -2.3-i0.25 = z3

y3=cosh(z3)    = 4.8806256579738629+i1.2214238973804283
w3=arccosh(y3) = 2.3+i0.25 = z1

y3=tanh(z3)    = -0.9824664000943306-i0.0094701778128957
w3=arctanh(y3) = -2.3000000000000002-i0.249999999999998 = z3

---------------
Fourth quadrant
---------------
z4=2.3-i0.25  = -z2

y4=sinh(z4)    = 4.783483618904492-i1.2462283322677304
w4=arcsinh(y4) = 2.3-i0.25 = z4

y4=cosh(z4)    = 4.8806256579738629-i1.2214238973804283
w4=arccosh(y4) = 2.3-i0.25 = z4

y4=tanh(z4)    = 0.9824664000943306-i0.0094701778128957
w4=arctanh(y4) = 2.3000000000000002-i0.249999999999998 = z4


[RuiLoureiro]

-------------------------------------------------------------------
about cos(w)=z and the inverse function w=arccos(z)
-------------------------------------------------------------------

Code Select Expand


By definition
                cos(w)= (e^iw+e^-iw)/2 = z   => w=arccos(z)

    From
                     e^iw+e^-iw
                 z=--------------   doing t=e^iw
                           2
    we get
                         1
                2z= t + ---  <=>   2z t = t^2 + 1  ( t<>0 )
                         t

                <=> t^2 -2z t + 1 = 0

                <=>   t= z + SQRT(z^2-1)        - positive solution I
                   or
                      t= z - SQRT(z^2-1)         - negative solution I

    From
            SQRT(z^2-1)= SQRT[(-1).(1-z^2)]= i SQRT(1-z^2)

    we have
                      t= z + i SQRT(1-z^2)      - positive solution II
                   or
                      t= z - i SQRT(1-z^2)      - negative solution II

    From
            t=e^iw  we get w=-i ln(t)

    Solutions:
   
            w=arccos(z)= -i ln(z +  SQRT(z^2-1))       - positive solution I
            w=arccos(z)= -i ln(z -  SQRT(z^2-1))       - negative solution I
            w=arccos(z)= -i ln(z +i SQRT(1-z^2))       - positive solution II
            w=arccos(z)= -i ln(z -i SQRT(1-z^2))       - negative solution II

    Because cos(w)=z and cos(-w)=z

            arccos(z)= w or -w


Here are some results

Code Select Expand


-----------------------------------
The calculator negative solution I
arccos(z) = -i * ln(z - sqrt(z^2-1)
-----------------------------------
z1=pi/4+i3 = 0.7853981633974483+i3 = -z3
y1=cos(z1)=  7.1189120679085491-i7.0837072942502341
w1=arccos(y1)= pi/4+i3 = z1

z2=-pi/4+i3 = -z4
y2=cos(z2)= 7.1189120679085491+i7.0837072942502341
w2=arccos(y2)= -pi/4+i3 = z2

z3=-pi/4-i3 = -z1
y3=cos(z3)= 7.1189120679085491-i7.0837072942502341
w3=arccos(y3)= pi/4+i3 = z1

z4=pi/4-i3 = -z2
y4=cos(z4)= 7.1189120679085491+i7.0837072942502341
w4=arccos(y4)= -pi/4+i3 = z2

------------------------------------
The calculator positive solution I
arccos(z) = -i * ln(z + sqrt(z^2-1))
------------------------------------
z1=pi/4+i3 = 0.7853981633974483+i3 = -z3
y1=cos(z1)=  7.1189120679085491-i7.0837072942502341
w1=arccos(y1)= -pi/4-i3 = z3

z2=-pi/4+i3 = -z4
y2=cos(z2)= 7.1189120679085491+i7.0837072942502341
w2=arccos(y2)= pi/4-i3 = z4

z3=-pi/4-i3 = -z1
y3=cos(z3)= 7.1189120679085491-i7.0837072942502341
w3=arccos(y3)= -pi/4-i3 = z3

z4=pi/4-i3 = -z2
y4=cos(z4)= 7.1189120679085491+i7.0837072942502341
w4=arccos(y4)= pi/4-i3 = z4

-------------------------------------
The calculator positive solution II
arccos(z) = -i * ln(z +i sqrt(1-z^2))
-------------------------------------
z1=pi/4+i3 = 0.7853981633974483+i3 = -z3
y1=cos(z1)=  7.1189120679085491-i7.0837072942502341
w1=arccos(y1)= pi/4+i3 = z1

z2=-pi/4+i3 = -z4
y2=cos(z2)= 7.1189120679085491+i7.0837072942502341
w2=arccos(y2)= pi/4-i3 = z4

z3=-pi/4-i3 = -z1
y3=cos(z3)= 7.1189120679085491-i7.0837072942502341
w3=arccos(y3)= pi/4+i3 = z1

z4=pi/4-i3 = -z2
y4=cos(z4)= 7.1189120679085491+i7.0837072942502341
w4=arccos(y4)= pi/4-i3 = z4

-------------------------------------
The calculator negative solution II
arccos(z) = -i * ln(z - i sqrt(1-z^2)
-------------------------------------
z1=pi/4+i3 = 0.7853981633974483+i3 = -z3
y1=cos(z1)=  7.1189120679085491-i7.0837072942502341
w1=arccos(y1)=-pi/4-i3 = z3

z2=-pi/4+i3 = -z4
y2=cos(z2)= 7.1189120679085491+i7.0837072942502341
w2=arccos(y2)= -pi/4+i3 = z2

z3=-pi/4-i3 = -z1
y3=cos(z3)= 7.1189120679085491-i7.0837072942502341
w3=arccos(y3)=-pi/4-i3 = z3

z4=pi/4-i3 = -z2
y4=cos(z4)= 7.1189120679085491+i7.0837072942502341
w4=arccos(y4)= -pi/4+i3 = z2

arccos(z) = +w or -w    because cos(w)=z and cos(-w)=z


From these results,
the calculator uses the positive solution II.

Code Select Expand


            w=arccos(z)= -i ln(z +i SQRT(1-z^2))    - positive solution II


Here some more results

Code Select Expand


--------------------------------------
The calculator negative solution
arccos(z) = -i * ln(z - sqrt(z^2-1))
--------------------------------------
x1=pi/4
x2=3 pi/4

z1=x1+ix1    = 0.7853981633974483+i0.7853981633974483
y1=cos(z1)   = 0.9366400694314301-i0.6142431274865956
w1=arccos(y1)= 0.7853981633974482+i0.7853981633974482  = z1

z2=-x1+ix1   = -0.7853981633974483+i0.7853981633974483
y2=cos(z2)   = 0.9366400694314301+i0.6142431274865956
w2=arccos(y2)= -0.7853981633974482+i0.7853981633974482 = z2

z3=-x1-ix1   = -0.7853981633974483-i0.7853981633974483
y3=cos(z3)   = 0.9366400694314301-i0.6142431274865956
w3=arccos(y3)=  0.7853981633974482+i0.7853981633974482 = z1

z4=x1-ix1    = 0.7853981633974483-i0.7853981633974483
y4=cos(z4)   = 0.9366400694314301+i0.6142431274865956
w4=arccos(y4)=-0.7853981633974482+i0.7853981633974482  = z2
------------------------------------------------------------
z1=x2+ix1    = 2.3561944901923449+i0.7853981633974483
y1=cos(z1)   = -0.93664006943143-i0.6142431274865956
w1=arccos(y1)= -2.3561944901923449-i0.7853981633974482 = z3

z2=-x2+ix1   = -2.3561944901923449+i0.7853981633974483
y2=cos(z2)   = -0.93664006943143+i0.6142431274865956
w2=arccos(y2)= 2.3561944901923449-i0.7853981633974482  = z4

z3=-x2-ix1   = -2.3561944901923449-i0.7853981633974483
y3=cos(z3)   = -0.93664006943143-i0.6142431274865956
w3=arccos(y3)= -2.3561944901923449-i0.7853981633974482 = z3

z4=x2-ix1    = 2.3561944901923449-i0.7853981633974483
y4=cos(z4)   = -0.93664006943143+i0.6142431274865956
w4=arccos(y4)= 2.3561944901923449-i0.7853981633974482  = z4

--------------------------------------
The calculator positive solution II
arccos(z) = -i * ln(z +i sqrt(1-z^2))
--------------------------------------
x1=  pi/4 = 0.7853981633974483
x2=3 pi/4 = 2.3561944901923449
x3=5 pi/4 = 3.9269908169872415
x4=7 pi/4 = 5.4977871437821381

z1=x1+ix1    = 0.7853981633974483+i0.7853981633974483
y1=cos(z1)   = 0.9366400694314301-i0.6142431274865956
w1=arccos(y1)= 0.7853981633974482+i0.7853981633974482 = z1

z2=-x1+ix1   = -0.7853981633974483+i0.7853981633974483
y2=cos(z2)   = 0.9366400694314301+i0.6142431274865956
w2=arccos(y2)= 0.7853981633974482-i0.7853981633974482 = z4

z3=-x1-ix1   = -0.7853981633974483-i0.7853981633974483
y3=cos(z3)   = 0.9366400694314301-i0.6142431274865956
w3=arccos(y3)= 0.7853981633974482+i0.7853981633974482 = z1

z4=x1-ix1    = 0.7853981633974483-i0.7853981633974483
y4=cos(z4)   = 0.9366400694314301+i0.6142431274865956
w4=arccos(y4)= 0.7853981633974482-i0.7853981633974482 = z4
------------------------------------------------------------
z1=x2+ix1    = 2.3561944901923449+i0.7853981633974483
y1=cos(z1)   = -0.93664006943143-i0.6142431274865956
w1=arccos(y1)=  2.3561944901923449+i0.7853981633974482  =z1

z2=-x2+ix1   = -2.3561944901923449+i0.7853981633974483
y2=cos(z2)   = -0.93664006943143+i0.6142431274865956
w2=arccos(y2)= 2.3561944901923449-i0.7853981633974482   =z4

z3=-x2-ix1   = -2.3561944901923449-i0.7853981633974483
y3=cos(z3)   = -0.93664006943143-i0.6142431274865956
w3=arccos(y3)= 2.3561944901923449+i0.7853981633974482  = z1

z4=x2-ix1    = 2.3561944901923449-i0.7853981633974483
y4=cos(z4)   = -0.93664006943143+i0.6142431274865956
w4=arccos(y4)= 2.3561944901923449-i0.7853981633974482  = z4
------------------------------------------------------------
z1=x1+ix2    = 0.7853981633974483+i2.3561944901923449
y1=cos(z1)   = 3.7637541395008347-i3.6967343997925613
w1=arccos(y1)= 0.7853981633974483+i2.3561944901923449   = z1

z2=-x1+ix2   = -0.7853981633974483+i2.3561944901923449
y2=cos(z2)   = 3.7637541395008347+i3.6967343997925613
w2=arccos(y2)= 0.7853981633974483-i2.3561944901923449   = z4

z3=-x1-ix2   = -0.7853981633974483-i2.3561944901923449
y3=cos(z3)   = 3.7637541395008347-i3.6967343997925613
w3=arccos(y3)= 0.7853981633974483+i2.3561944901923449   = z1

z4=x1-ix2    =  0.7853981633974483-i2.3561944901923449
y4=cos(z4)   =  3.7637541395008347+i3.6967343997925613
w4=arccos(y4)= 0.7853981633974483-i2.3561944901923449   = z4
------------------------------------------------------------
z1=x2+ix2    = 2.3561944901923449+i2.3561944901923449
y1=cos(z1)   = -3.7637541395008346-i3.6967343997925614
w1=arccos(y1)= 2.3561944901923449+i2.3561944901923449   = z1

z2=-x2+ix2   = -2.3561944901923449+i2.3561944901923449
y2=cos(z2)   = -3.7637541395008346+i3.6967343997925614
w2=arccos(y2)=  2.3561944901923449-i2.3561944901923449  = z4

z3=-x2-ix2   = -2.3561944901923449-i2.3561944901923449
y3=cos(z3)   = -3.7637541395008346-i3.6967343997925614
w3=arccos(y3)= 2.3561944901923449+i2.3561944901923449   = z1

z4=x2-ix2    = 2.3561944901923449-i2.3561944901923449
y4=cos(z4)   = -3.7637541395008346+i3.6967343997925614
w4=arccos(y4)= 2.3561944901923449-i2.3561944901923449   = z4
-------------------------------------------------------------
x1=  pi/4 = 0.7853981633974483
x2=3 pi/4 = 2.3561944901923449
x3=5 pi/4 = 3.9269908169872415
x4=7 pi/4 = 5.4977871437821381

z1=x3+ix1    =  3.9269908169872415+i0.7853981633974483
y1=cos(z1)   = -0.9366400694314301+i0.6142431274865956
w1=arccos(y1)=  2.356194490192345-i0.7853981633974482   = x2-ix1

z2=-x3+ix1   = -3.9269908169872415+i0.7853981633974483
y2=cos(z2)   = -0.9366400694314301-i0.6142431274865956
w2=arccos(y2)=  2.356194490192345+i0.7853981633974482   = x2+ix1

z3=-x3-ix1   = -3.9269908169872415-i0.7853981633974483
y3=cos(z3)   = -0.9366400694314301+i0.6142431274865956
w3=arccos(y3)=  2.356194490192345-i0.7853981633974482   = x2-ix1

z4=x3-ix1    =  3.9269908169872415-i0.7853981633974483
y4=cos(z4)   = -0.9366400694314301-i0.6142431274865956
w4=arccos(y4)=  2.356194490192345+i0.7853981633974482   = x2+ix1
------------------------------------------------------------------
z1=x3+ix2    =  3.9269908169872415+i2.3561944901923449
y1=cos(z1)   = -3.7637541395008349+i3.6967343997925611
w1=arccos(y1)=  2.356194490192345-i2.3561944901923449   = x2-ix2

z2=-x3+ix2   = -3.9269908169872415+i2.3561944901923449
y2=cos(z2)   = -3.7637541395008349-i3.6967343997925611
w2=arccos(y2)=  2.356194490192345+i2.3561944901923449   = x2+ix2

z3=-x3-ix2   = -3.9269908169872415-i2.3561944901923449
y3=cos(z3)   = -3.7637541395008349+i3.6967343997925611
w3=arccos(y3)=  2.356194490192345-i2.3561944901923449   = x2-ix2

z4=x3-ix2    =  3.9269908169872415-i2.3561944901923449
y4=cos(z4)   = -3.7637541395008349-i3.6967343997925611
w4=arccos(y4)=  2.356194490192345+i2.3561944901923449   = x2+ix2
; -------------------------------------------------------------------
x1=  pi/4 = 0.7853981633974483
x2=3 pi/4 = 2.3561944901923449
x3=5 pi/4 = 3.9269908169872415
x4=7 pi/4 = 5.4977871437821381

z1=x4+ix1    = 5.4977871437821381+i0.7853981633974483 = -z3
y1=cos(z1)   = 0.93664006943143+i0.6142431274865956
w1=arccos(y1)= 0.7853981633974483-i0.7853981633974482 = x1-ix1

z2=-x4+ix1   = -5.4977871437821381+i0.7853981633974483 = -z4
y2=cos(z2)   =  0.93664006943143-i0.6142431274865956
w2=arccos(y2)=  0.7853981633974483+i0.7853981633974482 = x1+ix1

z3=-x4-ix1   = -5.4977871437821381-i0.7853981633974483 = -z1
y3=cos(z3)   =  0.93664006943143+i0.6142431274865956
w3=arccos(y3)=  0.7853981633974483-i0.7853981633974482  = x1-ix1

z4=x4-ix1    = 5.4977871437821381-i0.7853981633974483  = -z2
y4=cos(z4)   = 0.93664006943143-i0.6142431274865956
w4=arccos(y4)= 0.7853981633974483+i0.7853981633974482  = x1+ix1
----------------------------------------------------------------
z1=x4+ix2    = 5.4977871437821381+i2.3561944901923449
y1=cos(z1)   = 3.7637541395008344+i3.6967343997925616
w1=arccos(y1)= 0.7853981633974483-i2.3561944901923449   =x1-ix2

z2=-x4+ix2   = -5.4977871437821381+i2.3561944901923449
y2=cos(z2)   = 3.7637541395008344-i3.6967343997925616
w2=arccos(y2)= 0.7853981633974483+i2.3561944901923449   =x1+ix2

z3=-x4-ix2   = -5.4977871437821381-i2.3561944901923449
y3=cos(z3)   = 3.7637541395008344+i3.6967343997925616
w3=arccos(y3)= 0.7853981633974483-i2.3561944901923449   =x1-ix2

z4=x4-ix2    = 5.4977871437821381-i2.3561944901923449
y4=cos(z4)   = 3.7637541395008344-i3.6967343997925616
w4=arccos(y4)= 0.7853981633974483+i2.3561944901923449   =x1+ix2
------------------------------
z1=x4+ix3    = 5.4977871437821381+i3.9269908169872415
y1=cos(z1)   = 17.951221702159904+i17.937289667062213
w1=arccos(y1)= 0.7853981633974484-i3.9269908169872415   =x1-ix3

z2=-x4+ix3   = -5.4977871437821381+i3.9269908169872415
y2=cos(z2)   = 17.951221702159904-i17.937289667062213
w2=arccos(y2)= 0.7853981633974484+i3.9269908169872415   =x1+ix3

z3=-x4-ix3   = -5.4977871437821381-i3.9269908169872415
y3=cos(z3)   =  17.951221702159904+i17.937289667062213
w3=arccos(y3)= 0.7853981633974484-i3.9269908169872415   =x1-ix3

z4=x4-ix3    = 5.4977871437821381-i3.9269908169872415
y4=cos(z4)   = 17.951221702159904-i17.937289667062213
w4=arccos(y4)= 0.7853981633974484+i3.9269908169872415   =x1+ix3
----------------------------------------------------------------
x1=  pi/4 = 0.7853981633974483
x2=3 pi/4 = 2.3561944901923449
x3=5 pi/4 = 3.9269908169872415
x4=7 pi/4 = 5.4977871437821381

z1=x4+ix4    = 5.4977871437821381+i5.4977871437821381
y1=cos(z1)   = 86.321884181857327+i86.318987996303526
w1=arccos(y1)=  0.7853981633974483-i5.4977871437821381  =x1-ix4

z2=-x4+ix4   = -5.4977871437821381+i5.4977871437821381
y2=cos(z2)   = 86.321884181857327-i86.318987996303526
w2=arccos(y2)= 0.7853981633974483+i5.4977871437821383   =x1+ix4

z3=-x4-ix4   = -5.4977871437821381-i5.4977871437821381
y3=cos(z3)   = 86.321884181857327+i86.318987996303526
w3=arccos(y3)= 0.7853981633974483-i5.4977871437821381   =x1-ix4

z4=x4-ix4    = 5.4977871437821381-i5.4977871437821381
y4=cos(z4)   = 86.321884181857327-i86.318987996303526
w4=arccos(y4)= 0.7853981633974483+i5.4977871437821383   =x1+ix4


[Gunther]

Hi Rui,

only one question: What algorithm do you use by solving a linear equation system?

Gunther

[RuiLoureiro]

Hi Rui,
only one question: What algorithm do you use by solving a linear equation system?
Gunther

Hi Gunther

Where is the linear equation system ?

Do you want to say, for example:
                      a X+ bY=c and dX+eY=f  ? etc.
I use matrices.
               

[dedndave]

Hi Rui,

the standard form for a linear equation is called the "slope-intercept" form:

Y=mX+b

m is the slope (=rise/run=(Y₂-Y₁)/(X₂-X₁))
b is the y-intercept (the value of Y when X=0)

many calculators let you enter 2 (X,Y) points (some allow m and b)
then, you can enter an X or Y value, and it will spit out the opposite

nearly any straight line can be described using the slope-intercept form
however, it gets a little tricky as the slope approaches infinity   :P
those are lines that graph stright up and down, like the line: X=0

[RuiLoureiro]

Hi Rui,

the standard form for a linear equation is called the "slope-intercept" form:

Y=mX+b

m is the slope (=rise/run=(Y₂-Y₁)/(X₂-X₁))
b is the y-intercept (the value of Y when X=0)

many calculators let you enter 2 (X,Y) points (some allow m and b)
then, you can enter an X or Y value, and it will spit out the opposite

nearly any straight line can be described using the slope-intercept form
however, it gets a little tricky as the slope approaches infinity   :P
those are lines that graph stright up and down, like the line: X=0

Hi Dave,  :t

              i know all about that
              Gunther asked for linear equation system

[Gunther]

Rui,

Do you want to say, for example:
                      a X+ bY=c and dX+eY=f  ? etc.
I use matrices.

that's clear. The direction of my question was: Do you use the Gaussian elimination?

Gunther