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Identical expressions
- 2sin²x-2sinx- one = zero
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-
Similar expressions
- 2sin²x+2sinx-1=0
- 2sin²x-2sinx+1=0
2sin²x-2sinx-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 2*sin (x) - 2*sin(x) - 1 = 0
$$\left(2 \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 1 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$- 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2$$
$$b = -2$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{2} + \frac{\sqrt{3}}{2}$$
$$w_{2} = \frac{1}{2} - \frac{\sqrt{3}}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} + \pi$$
$$\left(2 \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 1 = 0$$
transform
$$- 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2$$
$$b = -2$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (2) * (-1) = 12
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{2} + \frac{\sqrt{3}}{2}$$
$$w_{2} = \frac{1}{2} - \frac{\sqrt{3}}{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} + \pi$$
Rapid solution
[src]
/ ___\
|1 \/ 3 |
x1 = pi - asin|- - -----|
\2 2 /
$$x_{1} = \pi - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
/ ___\
|1 \/ 3 |
x2 = asin|- - -----|
\2 2 /
$$x_{2} = \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
/ / ___\\ / / ___\\
| |1 \/ 3 || | |1 \/ 3 ||
x3 = pi - re|asin|- + -----|| - I*im|asin|- + -----||
\ \2 2 // \ \2 2 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 \/ 3 || | |1 \/ 3 ||
x4 = I*im|asin|- + -----|| + re|asin|- + -----||
\ \2 2 // \ \2 2 //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}$$
x4 = re(asin(1/2 + sqrt(3)/2)) + i*im(asin(1/2 + sqrt(3)/2))
Sum and product of roots
[src]
sum
/ ___\ / ___\ / / ___\\ / / ___\\ / / ___\\ / / ___\\
|1 \/ 3 | |1 \/ 3 | | |1 \/ 3 || | |1 \/ 3 || | |1 \/ 3 || | |1 \/ 3 ||
pi - asin|- - -----| + asin|- - -----| + pi - re|asin|- + -----|| - I*im|asin|- + -----|| + I*im|asin|- + -----|| + re|asin|- + -----||
\2 2 / \2 2 / \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right) + \left(\left(\operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} + \left(\pi - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
product
/ / ___\\ / ___\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | |1 \/ 3 || |1 \/ 3 | | | |1 \/ 3 || | |1 \/ 3 ||| | | |1 \/ 3 || | |1 \/ 3 ||| |pi - asin|- - -----||*asin|- - -----|*|pi - re|asin|- + -----|| - I*im|asin|- + -----|||*|I*im|asin|- + -----|| + re|asin|- + -----||| \ \2 2 // \2 2 / \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$\left(\pi - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\right) \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right)$$
=
/ / ___\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / ___\ | |1 \/ 3 || | | |1 \/ 3 || | |1 \/ 3 ||| | | |1 \/ 3 || | |1 \/ 3 ||| |1 \/ 3 | -|pi - asin|- - -----||*|I*im|asin|- + -----|| + re|asin|- + -----|||*|-pi + I*im|asin|- + -----|| + re|asin|- + -----|||*asin|- - -----| \ \2 2 // \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 /// \2 2 /
$$- \left(\pi - \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{1}{2} + \frac{\sqrt{3}}{2} \right)}\right)}\right) \operatorname{asin}{\left(\frac{1}{2} - \frac{\sqrt{3}}{2} \right)}$$
-(pi - asin(1/2 - sqrt(3)/2))*(i*im(asin(1/2 + sqrt(3)/2)) + re(asin(1/2 + sqrt(3)/2)))*(-pi + i*im(asin(1/2 + sqrt(3)/2)) + re(asin(1/2 + sqrt(3)/2)))*asin(1/2 - sqrt(3)/2)
Numerical answer
[src]
x1 = -19.2242903542475
x2 = -34.182784756779
x3 = -53.0323406783177
x4 = -50.6402168901454
x5 = 3.51632708629853
x6 = 75.0234892534463
x7 = 116.613662615531
x8 = 87.5898598678055
x9 = -15.3332288352402
x10 = -88.339328733223
x11 = -75.7729581188638
x12 = -21.6164141424198
x13 = -31.7906609686067
x14 = -44.3570315829658
x15 = 41.215438929376
x16 = -2.76685822088105
x17 = 31.0411921031892
x18 = -103.297823135754
x19 = -46.7491553711382
x20 = -97.0146378285748
x21 = 47.4986242365556
x22 = -0.37473443270874
x23 = 62.4571186390871
x24 = 72.631365465274
x25 = 85.1977360796332
x26 = -69.4897728116842
x27 = 93.8730451749851
x28 = -56.923402197325
x29 = 16.0826977006577
x30 = -27.8995994495994
x31 = 5.90845087447085
x32 = -78.1650819070361
x33 = -6.65791973988833
x34 = 56.1739333319075
x35 = -82.0561434260434
x36 = 49.890748024728
x37 = 18.47482148883
x38 = 68.7403039462667
x39 = 37.3243774103688
x40 = 100.156230482165
x41 = -65.5987112926769
x42 = 22.3658830078373
x43 = -71.8818965998565
x44 = -25.5074756614271
x45 = -63.2065875045046
x46 = 9.79951239347812
x47 = 43.6075627175484
x48 = 12.1916361816504
x49 = 66.3481801580944
x50 = 53.7818095437352
x51 = 28.6490683150169
x52 = 60.0649948509148
x53 = -38.0738462757863
x54 = -84.4482672142157
x55 = 24.7580067960096
x56 = 91.4809213868127
x57 = -94.6225140404025
x58 = -12.9411050470679
x59 = 78.9145507724536
x60 = -59.3155259854973
x61 = -1372.50125518603
x62 = -115.864193750114
x63 = 97.7641066939923
x64 = -90.7314525213953
x65 = -40.4659700639586
x65 = -40.4659700639586