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Similar expressions
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sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)+sqrt(3)/2=0 equation
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The solution
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___
___ \/ 3
sin(x)*(2*sin(x) - 1) - \/ 3 *sin(x) + ----- = 0
2
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Detail solution
Given the equation
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - \sqrt{3} \sin{\left(x \right)} - \sin{\left(x \right)} + \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand the expression in the equation
$$w \left(2 w - 1\right) - \sqrt{3} w + \frac{\sqrt{3}}{2} = 0$$
We get the quadratic equation
$$2 w^{2} - \sqrt{3} w - w + \frac{\sqrt{3}}{2} = 0$$
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
$$a = 2$$
$$b = - \sqrt{3} - 1$$
$$c = \frac{\sqrt{3}}{2}$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
$$w_{2} = - \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - \sqrt{3} \sin{\left(x \right)} - \sin{\left(x \right)} + \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand the expression in the equation
$$w \left(2 w - 1\right) - \sqrt{3} w + \frac{\sqrt{3}}{2} = 0$$
We get the quadratic equation
$$2 w^{2} - \sqrt{3} w - w + \frac{\sqrt{3}}{2} = 0$$
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
$$a = 2$$
$$b = - \sqrt{3} - 1$$
$$c = \frac{\sqrt{3}}{2}$$
, then
D = b^2 - 4 * a * c =
(-1 - sqrt(3))^2 - 4 * (2) * (sqrt(3)/2) = (-1 - sqrt(3))^2 - 4*sqrt(3)
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
$$w_{2} = - \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
Rapid solution
[src]
pi
x1 = --
6
$$x_{1} = \frac{\pi}{6}$$
pi
x2 = --
3
$$x_{2} = \frac{\pi}{3}$$
2*pi
x3 = ----
3
$$x_{3} = \frac{2 \pi}{3}$$
5*pi
x4 = ----
6
$$x_{4} = \frac{5 \pi}{6}$$
x4 = 5*pi/6
Sum and product of roots
[src]
sum
pi pi 2*pi 5*pi -- + -- + ---- + ---- 6 3 3 6
$$\frac{5 \pi}{6} + \left(\left(\frac{\pi}{6} + \frac{\pi}{3}\right) + \frac{2 \pi}{3}\right)$$
=
2*pi
$$2 \pi$$
product
pi pi 2*pi 5*pi --*--*----*---- 6 3 3 6
$$\frac{5 \pi}{6} \frac{2 \pi}{3} \frac{\pi}{6} \frac{\pi}{3}$$
=
4 5*pi ----- 162
$$\frac{5 \pi^{4}}{162}$$
5*pi^4/162
Numerical answer
[src]
x1 = 57.0722665402146
x2 = 84.2994028713261
x3 = -72.7802298081635
x4 = -93.7241808320955
x5 = -110.479341651241
x6 = -73.3038285837618
x7 = -389.033890269536
x8 = 65.4498469497874
x9 = 90.5825881785057
x10 = -10.471975511966
x11 = 40.317105721069
x12 = 82.7286065445312
x13 = 38.7463093942741
x14 = 96.8657734856853
x15 = -68.5914396033772
x16 = 6.80678408277789
x17 = 69.6386371545737
x18 = 8.37758040957278
x19 = 63.8790506229925
x20 = 25.6563400043166
x21 = -37.1755130674792
x22 = 8.90117918517108
x23 = -18.3259571459405
x24 = -41.8879020478639
x25 = -5.75958653158129
x26 = -97.9129710368819
x27 = -29.3215314335047
x28 = -3.66519142918809
x29 = -24.60914245312
x30 = 59.1666616426078
x31 = -79.5870138909414
x32 = -49.7418836818384
x33 = 77.4926187885482
x34 = -48.1710873550435
x35 = 0.523598775598299
x36 = -16.2315620435473
x37 = -92.1533845053006
x38 = -17.8023583703422
x39 = 34.0339204138894
x40 = -35.081117965086
x41 = 52.8834763354282
x42 = -54.4542726622231
x43 = -35.6047167406843
x44 = 14.6607657167524
x45 = 27.7507351067098
x46 = -24.0855436775217
x47 = -62.3082542961976
x48 = 50.789081233035
x49 = 26.1799387799149
x50 = 2.61799387799149
x51 = -41.3643032722656
x52 = -9.94837673636768
x53 = 19.8967534727354
x54 = 78.0162175641465
x55 = -56.025068989018
x56 = 96.342174710087
x57 = -12.0427718387609
x58 = 88.4881930761125
x59 = -43.4586983746588
x60 = -74.8746249105567
x61 = 31.9395253114962
x62 = -173.834793498635
x63 = -66.497044500984
x64 = -30.8923277602996
x65 = -60.2138591938044
x66 = -79.0634151153431
x67 = 46.6002910282486
x68 = -61.7846555205993
x69 = 46.0766922526503
x70 = 63.3554518473942
x71 = 21.4675497995303
x72 = -85.870199198121
x73 = 52.3598775598299
x74 = -68.0678408277789
x75 = -47.6474885794452
x76 = 2.0943951023932
x77 = 82.2050077689329
x78 = -53.9306738866248
x79 = -87.4409955249159
x80 = -81.1578102177363
x81 = 19.3731546971371
x82 = 101.054563690472
x83 = 90.0589894029074
x84 = 76.4454212373516
x85 = 44.5058959258554
x86 = -28.7979326579064
x87 = 13.0899693899575
x88 = 15.1843644923507
x89 = 71.733032256967
x90 = 94.7713783832921
x91 = -100.007366139275
x92 = -85.3466004225227
x93 = 38.2227106186758
x94 = 70.162235930172
x95 = -22.5147473507269
x95 = -22.5147473507269