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2sin^2x+sinx-1=0 equation
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The solution
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[src]
2 2*sin (x) + sin(x) - 1 = 0
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$\sin{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2$$
$$b = 1$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$\sin{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$\left(2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2$$
$$b = 1$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (2) * (-1) = 9
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Rapid solution
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
pi
x2 = --
6
$$x_{2} = \frac{\pi}{6}$$
5*pi
x3 = ----
6
$$x_{3} = \frac{5 \pi}{6}$$
3*pi
x4 = ----
2
$$x_{4} = \frac{3 \pi}{2}$$
x4 = 3*pi/2
Sum and product of roots
[src]
sum
pi pi 5*pi 3*pi - -- + -- + ---- + ---- 2 6 6 2
$$\left(\left(- \frac{\pi}{2} + \frac{\pi}{6}\right) + \frac{5 \pi}{6}\right) + \frac{3 \pi}{2}$$
=
2*pi
$$2 \pi$$
product
-pi pi 5*pi 3*pi ----*--*----*---- 2 6 6 2
$$\frac{3 \pi}{2} \frac{5 \pi}{6} \cdot - \frac{\pi}{2} \frac{\pi}{6}$$
=
4 -5*pi ------ 48
$$- \frac{5 \pi^{4}}{48}$$
-5*pi^4/48
Numerical answer
[src]
x1 = -32.98672341235
x2 = -79.0634151153431
x3 = 73.8274274783337
x4 = 80.1106131458253
x5 = -35.081117965086
x6 = 4.71238877821279
x7 = 88.4881930761125
x8 = -56.025068989018
x9 = -47.6474885794452
x10 = 98.9601683847854
x11 = 67.5442422659503
x12 = 38.2227106186758
x13 = 92.6769826185806
x14 = 75.9218224617533
x15 = 78.0162175641465
x16 = 84.2994028713261
x17 = -26.7035373476123
x18 = 42.4115007297604
x19 = -51.8362786898924
x20 = -76.9690201780717
x21 = 92.6769830871924
x22 = 44.5058959258554
x23 = 71.733032256967
x24 = -49.7418836818384
x25 = -87.4409955249159
x26 = -41.3643032722656
x27 = -68.5914396033772
x28 = -100.007366139275
x29 = -76.9690198122422
x30 = 69.6386371545737
x31 = -16.2315620435473
x32 = 61.2610569380464
x33 = 86.393797888715
x34 = 34.0339204138894
x35 = -22.5147473507269
x36 = -66.497044500984
x37 = -20.4203520418601
x38 = -64.4026491963026
x39 = 52.8834763354282
x40 = 23.5619451122289
x41 = -95.8185758681551
x42 = -60.2138591938044
x43 = 90.5825881785057
x44 = -83.2522055292846
x45 = -45.5530935873709
x46 = -97.9129710368819
x47 = 2.61799387799149
x48 = 94.7713783832921
x49 = -43.4586983746588
x50 = -62.3082542961976
x51 = -7.85398149924071
x52 = 54.9778712411975
x53 = -89.5353907455655
x54 = -14.1371668400256
x55 = -93.7241808320955
x56 = 29.8451303193672
x57 = 40.317105721069
x58 = 48.6946859325274
x59 = -91.6297857297023
x60 = 17.2787597959772
x61 = -85.3466004225227
x62 = -1.57079642893127
x63 = -58.1194639999037
x64 = 27.7507351067098
x65 = -5.75958653158129
x66 = 46.6002910282486
x67 = -9.94837673636768
x68 = 10.9955740992967
x69 = 36.1283159916529
x70 = 63.3554518473942
x71 = -76.9690204511548
x72 = -53.9306738866248
x73 = 10.995574056153
x74 = -12.0427718387609
x75 = -95.8185760435073
x76 = 82.2050077689329
x77 = -39.2699083757319
x78 = 96.8657734856853
x79 = -32.9867230405965
x80 = 31.9395253114962
x81 = 0.523598775598299
x82 = -3.66519142918809
x83 = 8.90117918517108
x84 = -70.6858344924983
x85 = 25.6563400043166
x86 = -24.60914245312
x87 = -18.3259571459405
x88 = 19.3731546971371
x88 = 19.3731546971371
The graph