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Similar expressions
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sin^3x-sin^2x+sinx-1=0 equation
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The solution
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[src]
3 2 sin (x) - sin (x) + sin(x) - 1 = 0
$$\left(\left(\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) + \sin{\left(x \right)}\right) - 1 = 0$$
Detail solution
Given the equation
$$\left(\left(\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) + \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)} + \sin{\left(x \right)} - 1 = 0$$
$$\left(\left(\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) + \sin{\left(x \right)}\right) - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$w^{3} - w^{2} + w - 1 = 0$$
transform
$$\left(w + \left(\left(- w^{2} + \left(w^{3} - 1\right)\right) + 1\right)\right) - 1 = 0$$
or
$$\left(w + \left(\left(- w^{2} + \left(w^{3} - 1^{3}\right)\right) + 1^{2}\right)\right) - 1 = 0$$
$$\left(w - 1\right) + \left(- (w^{2} - 1^{2}) + \left(w^{3} - 1^{3}\right)\right) = 0$$
$$\left(w - 1\right) + \left(- (w - 1) \left(w + 1\right) + \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Take common factor -1 + w from the equation
we get:
$$\left(w - 1\right) \left(\left(- (w + 1) + \left(\left(w^{2} + w\right) + 1^{2}\right)\right) + 1\right) = 0$$
or
$$\left(w - 1\right) \left(w^{2} + 1\right) = 0$$
then:
$$w_{1} = 1$$
and also
we get the equation
$$w^{2} + 1 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$w_{2} = i$$
$$w_{3} = - i$$
The final answer for sin(x)^3 - sin(x)^2 + sin(x) - 1 = 0:
$$w_{1} = 1$$
$$w_{2} = i$$
$$w_{3} = - i$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(i \right)}$$
$$x_{2} = 2 \pi n + i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(w_{3} \right)}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(- i \right)}$$
$$x_{3} = 2 \pi n - i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$x_{5} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{5} = 2 \pi n + \pi - \operatorname{asin}{\left(i \right)}$$
$$x_{5} = 2 \pi n + \pi - i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{6} = 2 \pi n - \operatorname{asin}{\left(w_{3} \right)} + \pi$$
$$x_{6} = 2 \pi n + \pi - \operatorname{asin}{\left(- i \right)}$$
$$x_{6} = 2 \pi n + \pi + i \log{\left(1 + \sqrt{2} \right)}$$
$$\left(\left(\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) + \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)} + \sin{\left(x \right)} - 1 = 0$$
$$\left(\left(\sin^{3}{\left(x \right)} - \sin^{2}{\left(x \right)}\right) + \sin{\left(x \right)}\right) - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$w^{3} - w^{2} + w - 1 = 0$$
transform
$$\left(w + \left(\left(- w^{2} + \left(w^{3} - 1\right)\right) + 1\right)\right) - 1 = 0$$
or
$$\left(w + \left(\left(- w^{2} + \left(w^{3} - 1^{3}\right)\right) + 1^{2}\right)\right) - 1 = 0$$
$$\left(w - 1\right) + \left(- (w^{2} - 1^{2}) + \left(w^{3} - 1^{3}\right)\right) = 0$$
$$\left(w - 1\right) + \left(- (w - 1) \left(w + 1\right) + \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Take common factor -1 + w from the equation
we get:
$$\left(w - 1\right) \left(\left(- (w + 1) + \left(\left(w^{2} + w\right) + 1^{2}\right)\right) + 1\right) = 0$$
or
$$\left(w - 1\right) \left(w^{2} + 1\right) = 0$$
then:
$$w_{1} = 1$$
and also
we get the equation
$$w^{2} + 1 = 0$$
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:
$$w_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (1) = -4
Because D<0, then the equation
has no real roots,
but complex roots is exists.
w2 = (-b + sqrt(D)) / (2*a)
w3 = (-b - sqrt(D)) / (2*a)
or
$$w_{2} = i$$
$$w_{3} = - i$$
The final answer for sin(x)^3 - sin(x)^2 + sin(x) - 1 = 0:
$$w_{1} = 1$$
$$w_{2} = i$$
$$w_{3} = - i$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(i \right)}$$
$$x_{2} = 2 \pi n + i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(w_{3} \right)}$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(- i \right)}$$
$$x_{3} = 2 \pi n - i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$x_{5} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{5} = 2 \pi n + \pi - \operatorname{asin}{\left(i \right)}$$
$$x_{5} = 2 \pi n + \pi - i \log{\left(1 + \sqrt{2} \right)}$$
$$x_{6} = 2 \pi n - \operatorname{asin}{\left(w_{3} \right)} + \pi$$
$$x_{6} = 2 \pi n + \pi - \operatorname{asin}{\left(- i \right)}$$
$$x_{6} = 2 \pi n + \pi + i \log{\left(1 + \sqrt{2} \right)}$$
Sum and product of roots
[src]
sum
pi / ___\ / ___\ / ___\ / ___\ -- - I*log\1 + \/ 2 / + I*log\1 + \/ 2 / + pi - I*log\1 + \/ 2 / + pi + I*log\1 + \/ 2 / 2
$$\left(\left(\pi - i \log{\left(1 + \sqrt{2} \right)}\right) + \left(\left(\frac{\pi}{2} - i \log{\left(1 + \sqrt{2} \right)}\right) + i \log{\left(1 + \sqrt{2} \right)}\right)\right) + \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
5*pi ---- 2
$$\frac{5 \pi}{2}$$
product
pi / / ___\\ / ___\ / / ___\\ / / ___\\ --*\-I*log\1 + \/ 2 //*I*log\1 + \/ 2 /*\pi - I*log\1 + \/ 2 //*\pi + I*log\1 + \/ 2 // 2
$$i \log{\left(1 + \sqrt{2} \right)} \frac{\pi}{2} \left(- i \log{\left(1 + \sqrt{2} \right)}\right) \left(\pi - i \log{\left(1 + \sqrt{2} \right)}\right) \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
2/ ___\ / 2 2/ ___\\
pi*log \1 + \/ 2 /*\pi + log \1 + \/ 2 //
------------------------------------------
2
$$\frac{\pi \left(\log{\left(1 + \sqrt{2} \right)}^{2} + \pi^{2}\right) \log{\left(1 + \sqrt{2} \right)}^{2}}{2}$$
pi*log(1 + sqrt(2))^2*(pi^2 + log(1 + sqrt(2))^2)/2
Rapid solution
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
/ ___\ x2 = -I*log\1 + \/ 2 /
$$x_{2} = - i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x3 = I*log\1 + \/ 2 /
$$x_{3} = i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x4 = pi - I*log\1 + \/ 2 /
$$x_{4} = \pi - i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x5 = pi + I*log\1 + \/ 2 /
$$x_{5} = \pi + i \log{\left(1 + \sqrt{2} \right)}$$
x5 = pi + i*log(1 + sqrt(2))
Numerical answer
[src]
x1 = -36.1283158039284
x2 = 51.8362790565645
x3 = -86.3937977849336
x4 = 1.57079652201191
x5 = 26.7035372816499
x6 = 1.57079657706073
x7 = 70.6858344346853
x8 = -4.71238879216987
x9 = -36.1283154211008
x10 = 32.9867229326546
x11 = 64.4026490775763
x12 = 70.6858349602121
x13 = 89.5353908274953
x14 = 51.8362785129677
x15 = 7.853981355985
x16 = -42.4115010896035
x17 = -54.9778716025053
x18 = -17.2787598746948
x19 = -67.5442421645771
x20 = -10.9955742131839
x21 = -10.9955744631114
x22 = 76.9690200801297
x23 = -73.827427819046
x24 = -17.2787593704775
x25 = -48.6946856290592
x26 = 64.4026497976315
x27 = -10.9955745946061
x28 = -73.8274272802379
x29 = -86.3937982482645
x30 = 1.57079607124023
x31 = 26.7035373672887
x32 = 45.5530936749393
x33 = 64.4026493101472
x34 = -23.5619450066319
x35 = -67.5442423581302
x36 = 58.1194636195043
x37 = -67.5442416896433
x38 = 39.2699083535838
x39 = -61.2610569386363
x40 = 76.9690198280084
x41 = -61.2610570351664
x42 = -80.1106125809448
x43 = 7.85398173817329
x44 = 51.8362788969468
x45 = 39.2699081011645
x46 = 70.6858345208175
x47 = 95.8185760552689
x48 = -36.1283152411117
x49 = -48.6946861971157
x50 = -98.960168501045
x51 = -92.6769833494036
x52 = 83.2522054987836
x53 = 102.10176077096
x54 = 32.9867224653991
x55 = -48.6946859384688
x56 = -98.9601687518542
x57 = 20.420351933225
x58 = 26.7035377955129
x59 = -92.6769830853013
x60 = 14.1371670999119
x61 = 83.2522056660379
x62 = 39.2699085924612
x63 = 83.2522052475869
x64 = 58.1194645258928
x65 = -17.2787597858663
x66 = 76.9690195579451
x67 = -80.1106123883388
x68 = 20.4203521516071
x69 = -42.4115005721817
x70 = -42.4115006312809
x71 = -80.1106129666365
x72 = 7.85398190726701
x73 = -61.2610565374937
x74 = -4.71238850863593
x75 = 45.5530937286541
x76 = -54.9778713576574
x77 = -23.561945212061
x78 = 45.5530932340975
x79 = -92.6769827649663
x80 = -73.8274268897365
x81 = 89.5353903987143
x82 = 58.1194644467934
x83 = 89.5353908835478
x84 = 32.9867226821498
x85 = 95.8185762065316
x86 = -86.3937977199213
x87 = -29.845130097246
x88 = 20.4203526343256
x89 = -29.8451306752755
x90 = -4.71238904699061
x91 = -23.5619445281858
x92 = 14.1371664708223
x93 = 95.8185756670852
x94 = 14.1371673890647
x95 = -29.8451297493721
x96 = -54.977871607776
x96 = -54.977871607776