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- Equation:
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Equation 2(x-4)=1
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Equation 4*x^2-25=0
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Equation sin(x)^(2)-sin(x)=0
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Identical expressions
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Similar expressions
- sin(x)^(2)+sin(x)=0
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sin(x)^(2)-sin(x)=0 equation
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The solution
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[src]
2 sin (x) - sin(x) = 0
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
transform
$$\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
transform
$$\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (0) = 1
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
Sum and product of roots
[src]
sum
pi -- + pi 2
$$\frac{\pi}{2} + \pi$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
product
pi 0*--*pi 2
$$\pi 0 \frac{\pi}{2}$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
pi
x2 = --
2
$$x_{2} = \frac{\pi}{2}$$
x3 = pi
$$x_{3} = \pi$$
x3 = pi
Numerical answer
[src]
x1 = -50.2654824574367
x2 = 76.969019673036
x3 = 25.1327412287183
x4 = 18.8495559215388
x5 = -1564.51314148772
x6 = 94.2477796076938
x7 = 7.85398173796495
x8 = 65.9734457253857
x9 = -34.5575191894877
x10 = -61.2610569243204
x11 = 45.553093663481
x12 = -84.8230016469244
x13 = -15.707963267949
x14 = 81.6814089933346
x15 = 6.28318530717959
x16 = -4.7123888305818
x17 = -31.4159265358979
x18 = 39.2699080280542
x19 = 78.5398163397448
x20 = -87.9645943005142
x21 = 120.951318648179
x22 = -72.2566310325652
x23 = -86.3937977915432
x24 = -29.8451300972765
x25 = -37.6991118430775
x26 = -67.5442421642546
x27 = 20.42035215177
x28 = 14.1371670985871
x29 = 12.5663706143592
x30 = 58.1194647431527
x31 = -97.3893722612836
x32 = 72.2566310325652
x33 = 89.5353908137952
x34 = -80.1106125810393
x35 = -36.1283154212439
x36 = -6.28318530717959
x37 = 76.9690200976964
x38 = -9.42477796076938
x39 = -48.6946861243056
x40 = 62.8318530717959
x41 = 53.4070751110265
x42 = -25.1327412287183
x43 = 100.530964914873
x44 = 0.0
x45 = -65.9734457253857
x46 = -21.9911485751286
x47 = 97.3893722612836
x48 = -78.5398163397448
x49 = -10.9955739732138
x50 = 50.2654824574367
x51 = -23.5619450064001
x52 = 43.9822971502571
x53 = -53.4070751110265
x54 = -18.8495559215388
x55 = -54.9778709863297
x56 = 26.7035373768773
x57 = -62.8318530717959
x58 = 95.8185760548644
x59 = 15.707963267949
x60 = 28.2743338823081
x61 = 32.986723044911
x62 = 51.8362788966528
x63 = 21.9911485751286
x64 = 83.2522050600807
x65 = -10.9955747331165
x66 = 3.14159265358979
x67 = 37.6991118430775
x68 = 91.106186954104
x69 = -17.2787597741434
x70 = 39.2699086388565
x71 = 59.6902604182061
x72 = -28.2743338823081
x73 = -59.6902604182061
x74 = 69.1150383789755
x75 = -40.8407044966673
x76 = 56.5486677646163
x77 = 64.4026493102586
x78 = 34.5575191894877
x79 = 70.6858345286456
x80 = -69.1150383789755
x81 = -81.6814089933346
x82 = -92.6769832292373
x83 = -4.71238903613963
x84 = 9.42477796076938
x85 = 87.9645943005142
x86 = 32.9867223690379
x87 = -43.9822971502571
x88 = 83.2522058001693
x89 = -94.2477796076938
x90 = -75.398223686155
x91 = 47.1238898038469
x92 = 1.57079651244662
x93 = -54.9778717129156
x94 = -42.4115006392452
x95 = -73.8274272802392
x95 = -73.8274272802392
The graph