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Similar expressions
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sinx+(cosx)^2=0 equation
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The solution
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[src]
2 sin(x) + cos (x) = 0
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
transform
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1$$
$$b = 1$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
$$w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
transform
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1$$
$$b = 1$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-1) * (1) = 5
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
$$w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
Rapid solution
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/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
x1 = 2*re|atan|- - + ----- + --------------------|| + 2*I*im|atan|- - + ----- + --------------------||
\ \ 2 2 2 // \ \ 2 2 2 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
|1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
x2 = -2*atan|- + ----- + --------------------|
\2 2 2 /
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}$$
/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
| |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
x3 = - 2*re|atan|- - ----- + --------------------|| - 2*I*im|atan|- - ----- + --------------------||
\ \2 2 2 // \ \2 2 2 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
|1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
x4 = -2*atan|- + ----- - --------------------|
\2 2 2 /
$$x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
x4 = -2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + 1/2 + sqrt(5)/2)
Sum and product of roots
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sum
/ / ___________\\ / / ___________\\ / ___________\ / / ___________\\ / / ___________\\ / ___________\
| | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ |
| | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || |1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
2*re|atan|- - + ----- + --------------------|| + 2*I*im|atan|- - + ----- + --------------------|| - 2*atan|- + ----- + --------------------| + - 2*re|atan|- - ----- + --------------------|| - 2*I*im|atan|- - ----- + --------------------|| - 2*atan|- + ----- - --------------------|
\ \ 2 2 2 // \ \ 2 2 2 // \2 2 2 / \ \2 2 2 // \ \2 2 2 // \2 2 2 /
$$- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) + \left(- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right)\right)\right)$$
=
/ ___________\ / ___________\ / / ___________\\ / / ___________\\ / / ___________\\ / / ___________\\
| ___ ___ / ___ | | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ ||
|1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||
- 2*atan|- + ----- + --------------------| - 2*atan|- + ----- - --------------------| - 2*re|atan|- - ----- + --------------------|| + 2*re|atan|- - + ----- + --------------------|| - 2*I*im|atan|- - ----- + --------------------|| + 2*I*im|atan|- - + ----- + --------------------||
\2 2 2 / \2 2 2 / \ \2 2 2 // \ \ 2 2 2 // \ \2 2 2 // \ \ 2 2 2 //
$$- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}$$
product
/ / / ___________\\ / / ___________\\\ / ___________\ / / / ___________\\ / / ___________\\\ / ___________\ | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | | | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |2*re|atan|- - + ----- + --------------------|| + 2*I*im|atan|- - + ----- + --------------------|||*-2*atan|- + ----- + --------------------|*|- 2*re|atan|- - ----- + --------------------|| - 2*I*im|atan|- - ----- + --------------------|||*-2*atan|- + ----- - --------------------| \ \ \ 2 2 2 // \ \ 2 2 2 /// \2 2 2 / \ \ \2 2 2 // \ \2 2 2 /// \2 2 2 /
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}\right)$$
=
/ / / ___________\\ / / ___________\\\ / / / ___________\\ / / ___________\\\ / ___________\ / ___________\
| | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | ___ ___ / ___ |
| | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | |1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| | | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 || | | 1 \/ 5 \/ 2 *\/ 1 - \/ 5 ||| |1 \/ 5 \/ 2 *\/ 1 + \/ 5 | |1 \/ 5 \/ 2 *\/ 1 + \/ 5 |
-16*|I*im|atan|- - ----- + --------------------|| + re|atan|- - ----- + --------------------|||*|I*im|atan|- - + ----- + --------------------|| + re|atan|- - + ----- + --------------------|||*atan|- + ----- + --------------------|*atan|- + ----- - --------------------|
\ \ \2 2 2 // \ \2 2 2 /// \ \ \ 2 2 2 // \ \ 2 2 2 /// \2 2 2 / \2 2 2 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + \frac{\sqrt{2} \sqrt{1 - \sqrt{5}}}{2} \right)}\right)}\right) \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)} \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
-16*(i*im(atan(1/2 - sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)) + re(atan(1/2 - sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)))*(i*im(atan(-1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)) + re(atan(-1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2)))*atan(1/2 + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2)*atan(1/2 + sqrt(5)/2 - sqrt(2)*sqrt(1 + sqrt(5))/2)
Numerical answer
[src]
x1 = 43.3160577177646
x2 = -38.36535127557
x3 = -0.666239432492515
x4 = 99.8647254823809
x5 = -27.6080944498156
x6 = -65.3072062928931
x7 = 87.2983548680217
x8 = 54.073314543519
x9 = 47.7901292363394
x10 = -21.324909142636
x11 = 49.5992430249442
x12 = -63.4980925042884
x13 = -76.0644631186476
x14 = 10.0910173932619
x15 = -15.0417238354565
x16 = -57.2149071971088
x17 = -13.2326100468517
x18 = -88.6308337330067
x19 = 3.80783208608231
x20 = -8.75853852827686
x21 = 85.4892410794169
x22 = 81.0151695608421
x23 = 60.3564998506986
x24 = -32.0821659683904
x25 = 66.6396851578782
x26 = 13405.8420923001
x27 = 41.5069439291598
x28 = -82.3476484258271
x29 = 91.7724263865965
x30 = -71.5903916000727
x31 = -44.6485365827496
x32 = 11.9001311818667
x33 = -84.1567622144319
x34 = 72.9228704650578
x35 = -2.47535322109728
x36 = -52.740835678534
x37 = -94.9140190401863
x38 = -77.8735769072523
x39 = -33.8912797569952
x40 = -40.1744650641748
x41 = -25.7989806612109
x42 = -59.0240209857136
x43 = -69.781277811468
x44 = -19.5157953540313
x45 = -46.4576503713544
x46 = 5.61694587468707
x47 = 24.4665017962258
x48 = 98.0556116937761
x49 = 30.7496871034054
x50 = 18.1833164890462
x51 = -1917.03775812227
x52 = 16.3742027004415
x53 = -90.4399475216115
x54 = 62.1656136393034
x55 = 55.8824283321238
x56 = 74.7319842536625
x57 = 68.4487989464829
x58 = 22.6573880076211
x59 = 93.5815401752013
x60 = 28.9405733148007
x60 = 28.9405733148007