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sinx(2sinx-1)=0

sinx(2sinx-1)=0 equation

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Numerical solution:

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The solution

You have entered [src] 
sin(x)*(2*sin(x) - 1) = 0
(2sin⁡(x)−1)sin⁡(x)=0\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0
Simplify
Detail solution
Given the equation
(2sin⁡(x)−1)sin⁡(x)=0\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0
transform
(2sin⁡(x)−1)sin⁡(x)=0\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0
(2sin⁡(x)−1)sin⁡(x)=0\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0
Do replacement
w=sin⁡(x)w = \sin{\left(x \right)}
Expand the expression in the equation
w(2w−1)=0w \left(2 w - 1\right) = 0
We get the quadratic equation
2w2−w=02 w^{2} - w = 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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=2a = 2
b=−1b = -1
c=0c = 0
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (2) * (0) = 1

Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
w1=12w_{1} = \frac{1}{2}
w2=0w_{2} = 0
do backward replacement
sin⁡(x)=w\sin{\left(x \right)} = w
Given the equation
sin⁡(x)=w\sin{\left(x \right)} = w
- this is the simplest trigonometric equation
This equation is transformed to
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
Or
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, where n - is a integer
substitute w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(12)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x1=2πn+π6x_{1} = 2 \pi n + \frac{\pi}{6}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(0)x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}
x2=2πnx_{2} = 2 \pi n
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(12)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
x3=2πn+5π6x_{3} = 2 \pi n + \frac{5 \pi}{6}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(0)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi
x4=2πn+πx_{4} = 2 \pi n + \pi
The graph
0-80-60-40-2020406080-1001005-5
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = 0
x1=0x_{1} = 0 
     pi
x2 = --
     6
x2=π6x_{2} = \frac{\pi}{6} 
     5*pi
x3 = ----
      6
x3=5π6x_{3} = \frac{5 \pi}{6} 
x4 = pi
x4=πx_{4} = \pi 
x4 = pi
Sum and product of roots [src] 
sum
pi   5*pi     
-- + ---- + pi
6     6
(π6+5π6)+π\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \pi 
=
2*pi
2π2 \pi 
product
  pi 5*pi   
0*--*----*pi
  6   6
π5π6⋅0π6\pi \frac{5 \pi}{6} \cdot 0 \frac{\pi}{6} 
=
0
00 
0
Numerical answer [src] 
x1 = 25.1327412287183
x2 = 18.8495559215388
x3 = 94.2477796076938
x4 = -56.025068989018
x5 = -47.6474885794452
x6 = 65.9734457253857
x7 = -34.5575191894877
x8 = -84.8230016469244
x9 = 38.2227106186758
x10 = -15.707963267949
x11 = 75.9218224617533
x12 = 78.0162175641465
x13 = 6.28318530717959
x14 = 84.2994028713261
x15 = -31.4159265358979
x16 = -87.9645943005142
x17 = -72.2566310325652
x18 = -37.6991118430775
x19 = -143.989663289532
x20 = 128.281700021583
x21 = 12.5663706143592
x22 = -49.7418836818384
x23 = 72.2566310325652
x24 = -41.3643032722656
x25 = -68.5914396033772
x26 = -100.007366139275
x27 = 69.6386371545737
x28 = -106.290551446455
x29 = 34.0339204138894
x30 = 62.8318530717959
x31 = 53.4070751110265
x32 = -25.1327412287183
x33 = 100.530964914873
x34 = 0.0
x35 = -65.9734457253857
x36 = -21.9911485751286
x37 = 90.5825881785057
x38 = 97.3893722612836
x39 = -78.5398163397448
x40 = -74.8746249105567
x41 = -97.9129710368819
x42 = 2.61799387799149
x43 = 50.2654824574367
x44 = 43.9822971502571
x45 = -62.3082542961976
x46 = -53.4070751110265
x47 = -18.8495559215388
x48 = -62.8318530717959
x49 = -93.7241808320955
x50 = 15.707963267949
x51 = 28.2743338823081
x52 = 21.9911485751286
x53 = 40.317105721069
x54 = 3.14159265358979
x55 = -91.6297857297023
x56 = 91.106186954104
x57 = -85.3466004225227
x58 = 59.6902604182061
x59 = -28.2743338823081
x60 = -5.75958653158129
x61 = 46.6002910282486
x62 = -476.99848457005
x63 = -59.6902604182061
x64 = 69.1150383789755
x65 = -40.8407044966673
x66 = -9.94837673636768
x67 = 56.5486677646163
x68 = -69.1150383789755
x69 = -81.6814089933346
x70 = 50.789081233035
x71 = 9.42477796076938
x72 = -53.9306738866248
x73 = 87.9645943005142
x74 = -43.9822971502571
x75 = -12.0427718387609
x76 = 82.2050077689329
x77 = 96.8657734856853
x78 = 31.9395253114962
x79 = -75.398223686155
x80 = 47.1238898038469
x81 = -3.66519142918809
x82 = 25.6563400043166
x83 = -24.60914245312
x84 = -18.3259571459405
x85 = 19.3731546971371
x85 = 19.3731546971371
The graph
sinx(2sinx-1)=0 equation
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