Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+9=(x+9)^2
-
Equation cos(x)=-1
-
Equation 3*x+50=77
-
Equation 5-2(x-1)=4-x
- Express {x} in terms of y where:
- 10*x+5*y=-17
- 14*x+16*y=-3
- -19*x+1*y=0
- 10*x+18*y=6
-
Identical expressions
- tan5(x)=-sqrt(three)/ three
- tangent of 5(x) equally minus square root of (3) divide by 3
- tangent of 5(x) equally minus square root of (three) divide by three
- tan5(x)=-√(3)/3
- tan5x=-sqrt3/3
- tan5(x)=-sqrt(3) divide by 3
-
Similar expressions
- 2*tan(5*x)=-2
- tan(5*x-pi/3)=sqrt(3)/1
- tan5(x)=+sqrt(3)/3
- f*(x)=(tan(5*x))/x
- tan(5x)=t
- cos(x)=-(sqrt(3)/3)
tan5(x)=-sqrt(3)/3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
___
5 -\/ 3
tan (x) = -------
3
$$\tan^{5}{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Detail solution
Given the equation
$$\tan^{5}{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
transform
$$\tan^{5}{\left(x \right)} + \frac{\sqrt{3}}{3} = 0$$
$$\tan^{5}{\left(x \right)} - \frac{\left(-1\right) \sqrt{3}}{3} = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
Given the equation
$$w^{5} - \frac{\left(-1\right) \sqrt{3}}{3} = 0$$
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
$$\sqrt[5]{w^{5}} = \sqrt[5]{- \frac{\sqrt{3}}{3}}$$
or
$$w = \frac{\sqrt[10]{3} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{9}{10}} \cdot 3^{\frac{9}{10}} \sqrt[5]{-1} \cdot 3^{\frac{2}{5}}}{81}$$
Expand brackets in the right part
We get the answer: w = (-1)^(1/5)*3^(9/10)/3
All other 4 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{5} = - \frac{\sqrt{3}}{3}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{5} e^{5 i p} = - \frac{\sqrt{3}}{3}$$
where
$$r = \frac{3^{\frac{9}{10}}}{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{5 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1$$
so
$$\cos{\left(5 p \right)} = -1$$
and
$$\sin{\left(5 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{5} + \frac{\pi}{5}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{3^{\frac{9}{10}}}{3}$$
$$z_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}$$
$$z_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}$$
$$z_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}$$
$$z_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = - \frac{3^{\frac{9}{10}}}{3}$$
$$w_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}$$
$$w_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}$$
$$w_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}$$
$$w_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}$$
$$\tan^{5}{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
transform
$$\tan^{5}{\left(x \right)} + \frac{\sqrt{3}}{3} = 0$$
$$\tan^{5}{\left(x \right)} - \frac{\left(-1\right) \sqrt{3}}{3} = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
Given the equation
$$w^{5} - \frac{\left(-1\right) \sqrt{3}}{3} = 0$$
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
$$\sqrt[5]{w^{5}} = \sqrt[5]{- \frac{\sqrt{3}}{3}}$$
or
$$w = \frac{\sqrt[10]{3} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{9}{10}} \cdot 3^{\frac{9}{10}} \sqrt[5]{-1} \cdot 3^{\frac{2}{5}}}{81}$$
Expand brackets in the right part
w = 3^1/10*3^4/5*3^4/5*3^9/10*3^9/10-1^1/5*3^2/5)/81
We get the answer: w = (-1)^(1/5)*3^(9/10)/3
All other 4 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{5} = - \frac{\sqrt{3}}{3}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{5} e^{5 i p} = - \frac{\sqrt{3}}{3}$$
where
$$r = \frac{3^{\frac{9}{10}}}{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{5 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1$$
so
$$\cos{\left(5 p \right)} = -1$$
and
$$\sin{\left(5 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{5} + \frac{\pi}{5}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{3^{\frac{9}{10}}}{3}$$
$$z_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}$$
$$z_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}$$
$$z_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}$$
$$z_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = - \frac{3^{\frac{9}{10}}}{3}$$
$$w_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}$$
$$w_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}$$
$$w_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}$$
$$w_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}$$
Rapid solution
[src]
/ 9/10\
|3 |
x1 = -atan|-----|
\ 3 /
$$x_{1} = - \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)}$$
/ / ___________ ___________\\ / / ___________ ___________\\
| | / ___ / ___ || | | / ___ / ___ ||
| | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 ||
| | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- ||
| | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 ||
x2 = - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------||
\ \ 12 12 6 6 // \ \ 12 12 6 6 //
$$x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)}$$
/ / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\
| | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ ||
| | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 ||
| | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- ||
| | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 ||
x3 = - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------||
\ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}$$
/ / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\
| | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ ||
| | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 ||
| | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- ||
| |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 ||
x4 = I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------||
\ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 //
$$x_{4} = \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}$$
/ / ___________\\ / / ___________\\
| | / ___ || | | / ___ ||
| | 9/10 / 5 \/ 5 || | | 9/10 / 5 \/ 5 ||
| | 9/10 9/10 ___ I*3 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- ||
| |3 3 *\/ 5 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 ||
x5 = I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------||
\ \ 12 12 3 // \ \ 12 12 3 //
$$x_{5} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)}$$
x5 = re(atan(3^(9/10)/12 + 3^(9/10)*sqrt(5)/12 + 3^(9/10)*i*sqrt(5/8 - sqrt(5)/8)/3)) + i*im(atan(3^(9/10)/12 + 3^(9/10)*sqrt(5)/12 + 3^(9/10)*i*sqrt(5/8 - sqrt(5)/8)/3))
Sum and product of roots
[src]
sum
/ / ___________ ___________\\ / / ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________\\ / / ___________\\
| | / ___ / ___ || | | / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ || | | / ___ ||
| | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 || | | 9/10 / 5 \/ 5 ||
/ 9/10\ | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 9/10 ___ I*3 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- ||
|3 | | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 ||
- atan|-----| + - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|| + - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------||
\ 3 / \ \ 12 12 6 6 // \ \ 12 12 6 6 // \ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 // \ \ 12 12 3 // \ \ 12 12 3 //
$$\left(\left(\left(- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)}\right)$$
=
/ / ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\
| | / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ ||
| | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 ||
/ 9/10\ | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 9/10 ___ I*3 * / - - ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 9/10 ___ I*3 * / - - ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- ||
|3 | | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 ||
- atan|-----| - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + I*im|atan|----- + ----------- + ------------------------|| + I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + re|atan|----- + ----------- + ------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------||
\ 3 / \ \ 12 12 6 6 // \ \ 12 3 12 12 12 12 // \ \ 12 12 3 // \ \ 12 3 12 12 12 12 // \ \ 12 12 6 6 // \ \ 12 3 12 12 12 12 // \ \ 12 12 3 // \ \ 12 3 12 12 12 12 //
$$- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}$$
product
/ / / ___________ ___________\\ / / ___________ ___________\\\ / / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\\ / / / ___________ ___________ ___________ ___________ ___________ ___________\\ / / ___________ ___________ ___________ ___________ ___________ ___________\\\ / / / ___________\\ / / ___________\\\
| | | / ___ / ___ || | | / ___ / ___ ||| | | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ ||| | | | / ___ / ___ / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ / ___ / ___ ||| | | | / ___ || | | / ___ |||
| | | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 ||| | | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 ||| | | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 || | | 9/10 / 5 \/ 5 / 5 \/ 5 9/10 / 5 \/ 5 9/10 / 5 \/ 5 9/10 ___ / 5 \/ 5 9/10 ___ / 5 \/ 5 ||| | | | 9/10 / 5 \/ 5 || | | 9/10 / 5 \/ 5 |||
/ 9/10\ | | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- I*3 *\/ 5 * / - - ----- ||| | | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- ||| | | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- || | | 9/10 3 * / - - ----- * / - + ----- I*3 * / - - ----- I*3 * / - + ----- I*3 *\/ 5 * / - - ----- I*3 *\/ 5 * / - + ----- ||| | | | 9/10 9/10 ___ I*3 * / - - ----- || | | 9/10 9/10 ___ I*3 * / - - ----- |||
|3 | | | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 || | | 3 3 *\/ 5 \/ 8 8 \/ 8 8 ||| | | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | | 3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 ||| | | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 || | |3 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 ||| | | |3 3 *\/ 5 \/ 8 8 || | |3 3 *\/ 5 \/ 8 8 |||
-atan|-----|*|- re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|||*|- re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|||*|I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|||*|I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------|||
\ 3 / \ \ \ 12 12 6 6 // \ \ 12 12 6 6 /// \ \ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 /// \ \ \ 12 3 12 12 12 12 // \ \ 12 3 12 12 12 12 /// \ \ \ 12 12 3 // \ \ 12 12 3 ///
$$\left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)}\right) \left(- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)}\right)$$
=
/ / / / ___________\\\ / / / ___________\\\\ / / / / ___________ ___________\\\ / / / ___________ ___________\\\\ / / / / ___________ ___________ ___________ ___________\\\ / / / ___________ ___________ ___________ ___________\\\\ / / / / ___________ ___________ ___________ ___________\\\ / / / ___________ ___________ ___________ ___________\\\\ | | | 9/10 | ___ ___ / ___ ||| | | 9/10 | ___ ___ / ___ |||| | | | 9/10 | ___ ___ / ___ ____ / ___ ||| | | 9/10 | ___ ___ / ___ ____ / ___ |||| | | | 9/10 | ___ ___ / ___ ___ / ___ ____ / ___ ____ / ___ ||| | | 9/10 | ___ ___ / ___ ___ / ___ ____ / ___ ____ / ___ |||| | | | 9/10 | ___ ___ / ___ ____ / ___ ____ / ___ ___ / ___ ||| | | 9/10 | ___ ___ / ___ ____ / ___ ____ / ___ ___ / ___ |||| / 9/10\ | | |3 *\1 + \/ 5 + I*\/ 2 *\/ 5 - \/ 5 /|| | |3 *\1 + \/ 5 + I*\/ 2 *\/ 5 - \/ 5 /||| | | |3 *\-2 + 2*\/ 5 + I*\/ 2 *\/ 5 - \/ 5 + I*\/ 10 *\/ 5 - \/ 5 /|| | |3 *\-2 + 2*\/ 5 + I*\/ 2 *\/ 5 - \/ 5 + I*\/ 10 *\/ 5 - \/ 5 /||| | | |3 *\-4 - 4*\/ 5 + I*\/ 2 *\/ 5 + \/ 5 + I*\/ 2 *\/ 5 - \/ 5 + I*\/ 10 *\/ 5 + \/ 5 - I*\/ 10 *\/ 5 - \/ 5 /|| | |3 *\-4 - 4*\/ 5 + I*\/ 2 *\/ 5 + \/ 5 + I*\/ 2 *\/ 5 - \/ 5 + I*\/ 10 *\/ 5 + \/ 5 - I*\/ 10 *\/ 5 - \/ 5 /||| | | |3 *\4 - 4*\/ 5 + I*\/ 2 *\/ 5 + \/ 5 + I*\/ 10 *\/ 5 + \/ 5 + I*\/ 10 *\/ 5 - \/ 5 - I*\/ 2 *\/ 5 - \/ 5 /|| | |3 *\4 - 4*\/ 5 + I*\/ 2 *\/ 5 + \/ 5 + I*\/ 10 *\/ 5 + \/ 5 + I*\/ 10 *\/ 5 - \/ 5 - I*\/ 2 *\/ 5 - \/ 5 /||| |3 | -|I*im|atan|------------------------------------------|| + re|atan|------------------------------------------|||*|I*im|atan|-----------------------------------------------------------------------|| + re|atan|-----------------------------------------------------------------------|||*|I*im|atan|--------------------------------------------------------------------------------------------------------------------------|| + re|atan|--------------------------------------------------------------------------------------------------------------------------|||*|I*im|atan|-------------------------------------------------------------------------------------------------------------------------|| + re|atan|-------------------------------------------------------------------------------------------------------------------------|||*atan|-----| \ \ \ 12 // \ \ 12 /// \ \ \ 24 // \ \ 24 /// \ \ \ 48 // \ \ 48 /// \ \ \ 48 // \ \ 48 /// \ 3 /
$$- \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(1 + \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}}\right)}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(1 + \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}}\right)}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(-2 + 2 \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{5 - \sqrt{5}}\right)}{24} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(-2 + 2 \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{5 - \sqrt{5}}\right)}{24} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} - 4 - \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} - 4 - \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} + 4 - \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} + 4 - \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)}\right) \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)}$$
-(i*im(atan(3^(9/10)*(1 + sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)))/12)) + re(atan(3^(9/10)*(1 + sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)))/12)))*(i*im(atan(3^(9/10)*(-2 + 2*sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)))/24)) + re(atan(3^(9/10)*(-2 + 2*sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)))/24)))*(i*im(atan(3^(9/10)*(-4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) - i*sqrt(10)*sqrt(5 - sqrt(5)))/48)) + re(atan(3^(9/10)*(-4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) - i*sqrt(10)*sqrt(5 - sqrt(5)))/48)))*(i*im(atan(3^(9/10)*(4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)) - i*sqrt(2)*sqrt(5 - sqrt(5)))/48)) + re(atan(3^(9/10)*(4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)) - i*sqrt(2)*sqrt(5 - sqrt(5)))/48)))*atan(3^(9/10)/3)
Numerical answer
[src]
x1 = 178322.35162535
x2 = -123.252691204132
x3 = 14.9773855538189
x4 = -35.2880969036177
x5 = -101.261542629003
x6 = 18.1189782074087
x7 = -79.2703940538749
x8 = -25.8633189428484
x9 = 46.3933120897169
x10 = 2.41101493945977
x11 = -11447607863.8933
x12 = 84.0924239327944
x13 = -63.5624307859259
x14 = -44.7128748643871
x15 = -899.226076640811
x16 = 96.6587945471536
x17 = -33625.1967490857
x18 = -66.7040234395157
x19 = 58.9596827040761
x20 = -468.827883099009
x21 = -41.5712822107973
x22 = 62.1012753576658
x23 = -10848.6500105597
x24 = -7.01376302130961
x25 = 17931.4802889764
x26 = -57.2792454787463
x27 = -47.8544675179769
x28 = 52.6764973968965
x29 = -22.7217262892586
x30 = -69.8456160931055
x31 = 36.9685341289475
x32 = 40.1101267825373
x33 = -91.836764668234
x34 = 74.667645972025
x35 = -19.5801336356688
x36 = 68.3844606648454
x37 = 80.9508312792046
x38 = 43.2517194361271
x39 = -955.774744405427
x40 = 3480.15408246336
x41 = -305.46506511234
x42 = -3396.7922362447
x43 = 30.6853488217679
x44 = 8.69420024663936
x45 = -85.5535793610544
x46 = 24.4021635145883
x47 = 39187.4961831649
x48 = 90.375609239974
x49 = -1569405.33182707
x50 = -3.87217036771981
x51 = -13.2969483284892
x52 = -5297621.29843638
x53 = -32.146504250028
x53 = -32.146504250028