2cos^2x+5sinx=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
5 sin ( x ) + 2 cos 2 ( x ) = 0 5 \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0 5 sin ( x ) + 2 cos 2 ( x ) = 0 transform
5 sin ( x ) + cos ( 2 x ) + 1 = 0 5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0 5 sin ( x ) + cos ( 2 x ) + 1 = 0 − 2 sin 2 ( x ) + 5 sin ( x ) + 2 = 0 - 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 2 = 0 − 2 sin 2 ( x ) + 5 sin ( x ) + 2 = 0 Do replacement
w = sin ( x ) w = \sin{\left(x \right)} w = sin ( x ) This equation is of the form
a*w^2 + b*w + c = 0 A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
w 1 = D − b 2 a w_{1} = \frac{\sqrt{D} - b}{2 a} w 1 = 2 a D − b w 2 = − D − b 2 a w_{2} = \frac{- \sqrt{D} - b}{2 a} w 2 = 2 a − D − b where D = b^2 - 4*a*c - it is the discriminant.
Because
a = − 2 a = -2 a = − 2 b = 5 b = 5 b = 5 c = 2 c = 2 c = 2 , then
D = b^2 - 4 * a * c = (5)^2 - 4 * (-2) * (2) = 41 Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a) w2 = (-b - sqrt(D)) / (2*a) or
w 1 = 5 4 − 41 4 w_{1} = \frac{5}{4} - \frac{\sqrt{41}}{4} w 1 = 4 5 − 4 41 w 2 = 5 4 + 41 4 w_{2} = \frac{5}{4} + \frac{\sqrt{41}}{4} w 2 = 4 5 + 4 41 do backward replacement
sin ( x ) = w \sin{\left(x \right)} = w sin ( x ) = w Given the equation
sin ( x ) = w \sin{\left(x \right)} = w sin ( x ) = w - this is the simplest trigonometric equation
This equation is transformed to
x = 2 π n + asin ( w ) x = 2 \pi n + \operatorname{asin}{\left(w \right)} x = 2 πn + asin ( w ) x = 2 π n − asin ( w ) + π x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi x = 2 πn − asin ( w ) + π Or
x = 2 π n + asin ( w ) x = 2 \pi n + \operatorname{asin}{\left(w \right)} x = 2 πn + asin ( w ) x = 2 π n − asin ( w ) + π x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi x = 2 πn − asin ( w ) + π , where n - is a integer
substitute w:
x 1 = 2 π n + asin ( w 1 ) x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)} x 1 = 2 πn + asin ( w 1 ) x 1 = 2 π n + asin ( 5 4 − 41 4 ) x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} x 1 = 2 πn + asin ( 4 5 − 4 41 ) x 1 = 2 π n + asin ( 5 4 − 41 4 ) x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} x 1 = 2 πn + asin ( 4 5 − 4 41 ) x 2 = 2 π n + asin ( w 2 ) x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)} x 2 = 2 πn + asin ( w 2 ) x 2 = 2 π n + asin ( 5 4 + 41 4 ) x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)} x 2 = 2 πn + asin ( 4 5 + 4 41 ) x 2 = 2 π n + asin ( 5 4 + 41 4 ) x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)} x 2 = 2 πn + asin ( 4 5 + 4 41 ) x 3 = 2 π n − asin ( w 1 ) + π x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi x 3 = 2 πn − asin ( w 1 ) + π x 3 = 2 π n − asin ( 5 4 − 41 4 ) + π x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi x 3 = 2 πn − asin ( 4 5 − 4 41 ) + π x 3 = 2 π n − asin ( 5 4 − 41 4 ) + π x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi x 3 = 2 πn − asin ( 4 5 − 4 41 ) + π x 4 = 2 π n − asin ( w 2 ) + π x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi x 4 = 2 πn − asin ( w 2 ) + π x 4 = 2 π n + π − asin ( 5 4 + 41 4 ) x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)} x 4 = 2 πn + π − asin ( 4 5 + 4 41 ) x 4 = 2 π n + π − asin ( 5 4 + 41 4 ) x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)} x 4 = 2 πn + π − asin ( 4 5 + 4 41 )
The graph
0 -80 -60 -40 -20 20 40 60 80 -100 100 -10 10
/ / ____________\\ / / ____________\\
| | ____ ____ / ____ || | | ____ ____ / ____ ||
| | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
x1 = 2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------||
\ \ 4 4 4 // \ \ 4 4 4 //
x 1 = 2 re ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) + 2 i im ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} x 1 = 2 re ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) + 2 i im ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) )
/ ____________\
| ____ ____ / ____ |
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
x2 = -2*atan|- + ------ + ----------------------|
\4 4 4 /
x 2 = − 2 atan ( 5 4 + 41 4 + 10 5 + 41 4 ) x_{2} = - 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} x 2 = − 2 atan ( 4 5 + 4 41 + 4 10 5 + 41 )
/ / ____________\\ / / ____________\\
| | ____ ____ / ____ || | | ____ ____ / ____ ||
| |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
x3 = - 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------||
\ \4 4 4 // \ \4 4 4 //
x 3 = − 2 re ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) − 2 i im ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} x 3 = − 2 re ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) − 2 i im ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) )
/ ____________\
| ____ ____ / ____ |
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
x4 = -2*atan|- + ------ - ----------------------|
\4 4 4 /
x 4 = − 2 atan ( − 10 5 + 41 4 + 5 4 + 41 4 ) x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} x 4 = − 2 atan ( − 4 10 5 + 41 + 4 5 + 4 41 )
x4 = -2*atan(-sqrt(10)*sqrt(5 + sqrt(41))/4 + 5/4 + sqrt(41)/4)
Sum and product of roots
[src]
/ / ____________\\ / / ____________\\ / ____________\ / / ____________\\ / / ____________\\ / ____________\
| | ____ ____ / ____ || | | ____ ____ / ____ || | ____ ____ / ____ | | | ____ ____ / ____ || | | ____ ____ / ____ || | ____ ____ / ____ |
| | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || |5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------|| - 2*atan|- + ------ + ----------------------| + - 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|| - 2*atan|- + ------ - ----------------------|
\ \ 4 4 4 // \ \ 4 4 4 // \4 4 4 / \ \4 4 4 // \ \4 4 4 // \4 4 4 /
− 2 atan ( − 10 5 + 41 4 + 5 4 + 41 4 ) + ( ( − 2 re ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) − 2 i im ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) ) + ( − 2 atan ( 5 4 + 41 4 + 10 5 + 41 4 ) + ( 2 re ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) + 2 i im ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) ) ) ) - 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) + \left(- 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right)\right)\right) − 2 atan ( − 4 10 5 + 41 + 4 5 + 4 41 ) + ( ( − 2 re ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) − 2 i im ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) ) + ( − 2 atan ( 4 5 + 4 41 + 4 10 5 + 41 ) + ( 2 re ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) + 2 i im ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) ) ) )
/ ____________\ / ____________\ / / ____________\\ / / ____________\\ / / ____________\\ / / ____________\\
| ____ ____ / ____ | | ____ ____ / ____ | | | ____ ____ / ____ || | | ____ ____ / ____ || | | ____ ____ / ____ || | | ____ ____ / ____ ||
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 | |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
- 2*atan|- + ------ - ----------------------| - 2*atan|- + ------ + ----------------------| - 2*re|atan|- - ------ + ----------------------|| + 2*re|atan|- - + ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------||
\4 4 4 / \4 4 4 / \ \4 4 4 // \ \ 4 4 4 // \ \4 4 4 // \ \ 4 4 4 //
− 2 atan ( 5 4 + 41 4 + 10 5 + 41 4 ) − 2 atan ( − 10 5 + 41 4 + 5 4 + 41 4 ) − 2 re ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) + 2 re ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) − 2 i im ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) + 2 i im ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) - 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} - 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} − 2 atan ( 4 5 + 4 41 + 4 10 5 + 41 ) − 2 atan ( − 4 10 5 + 41 + 4 5 + 4 41 ) − 2 re ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) + 2 re ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) − 2 i im ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) + 2 i im ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) )
/ / / ____________\\ / / ____________\\\ / ____________\ / / / ____________\\ / / ____________\\\ / ____________\
| | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ | | | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ |
| | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
|2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------|||*-2*atan|- + ------ + ----------------------|*|- 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|||*-2*atan|- + ------ - ----------------------|
\ \ \ 4 4 4 // \ \ 4 4 4 /// \4 4 4 / \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 /
( 2 re ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) + 2 i im ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) ) ( − 2 atan ( 5 4 + 41 4 + 10 5 + 41 4 ) ) ( − 2 re ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) − 2 i im ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) ) ( − 2 atan ( − 10 5 + 41 4 + 5 4 + 41 4 ) ) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)}\right) ( 2 re ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) + 2 i im ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) ) ( − 2 atan ( 4 5 + 4 41 + 4 10 5 + 41 ) ) ( − 2 re ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) − 2 i im ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) ) ( − 2 atan ( − 4 10 5 + 41 + 4 5 + 4 41 ) )
/ / / ____________\\ / / ____________\\\ / / / ____________\\ / / ____________\\\ / ____________\ / ____________\
| | | ____ ____ / ____ || | | ____ ____ / ____ ||| | | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ | | ____ ____ / ____ |
| | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | |5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
-16*|I*im|atan|- - + ------ + ----------------------|| + re|atan|- - + ------ + ----------------------|||*|I*im|atan|- - ------ + ----------------------|| + re|atan|- - ------ + ----------------------|||*atan|- + ------ - ----------------------|*atan|- + ------ + ----------------------|
\ \ \ 4 4 4 // \ \ 4 4 4 /// \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 / \4 4 4 /
− 16 ( re ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) + i im ( atan ( − 5 4 + 41 4 + 10 5 − 41 4 ) ) ) ( re ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) + i im ( atan ( − 41 4 + 5 4 + 10 5 − 41 4 ) ) ) atan ( 5 4 + 41 4 + 10 5 + 41 4 ) atan ( − 10 5 + 41 4 + 5 4 + 41 4 ) - 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} − 16 ( re ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) + i im ( atan ( − 4 5 + 4 41 + 4 10 5 − 41 ) ) ) ( re ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) + i im ( atan ( − 4 41 + 4 5 + 4 10 5 − 41 ) ) ) atan ( 4 5 + 4 41 + 4 10 5 + 41 ) atan ( − 4 10 5 + 41 + 4 5 + 4 41 )
-16*(i*im(atan(-5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)) + re(atan(-5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)))*(i*im(atan(5/4 - sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)) + re(atan(5/4 - sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)))*atan(5/4 + sqrt(41)/4 - sqrt(10)*sqrt(5 + sqrt(41))/4)*atan(5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 + sqrt(41))/4)