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- Equation:
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Equation 8-5(8+3x)=13
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Equation 2x^2+13-13x=x+1
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Equation x^4-8x^2+16=0
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Equation tan(pi*(4x-18)/6)=sqrt(3)/3
- Express {x} in terms of y where:
- 4*x-16*y=6
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
- 10*x-1*y=10
- 15*x-17*y=19
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Identical expressions
- 2cos^3x-2cosx-sin^2x= zero
- 2 co sinus of e of cubed x minus 2 co sinus of e of x minus sinus of squared x equally 0
- 2 co sinus of e of cubed x minus 2 co sinus of e of x minus sinus of squared x equally zero
- 2cos3x-2cosx-sin2x=0
- 2cos³x-2cosx-sin²x=0
- 2cos to the power of 3x-2cosx-sin to the power of 2x=0
- 2cos^3x-2cosx-sin^2x=O
-
Similar expressions
- 2cos^3x-2cosx+sin^2x=0
- 2cos^3x+2cosx-sin^2x=0
2cos^3x-2cosx-sin^2x=0 equation
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The solution
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[src]
3 2 2*cos (x) - 2*cos(x) - sin (x) = 0
$$\left(2 \cos^{3}{\left(x \right)} - 2 \cos{\left(x \right)}\right) - \sin^{2}{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\left(2 \cos^{3}{\left(x \right)} - 2 \cos{\left(x \right)}\right) - \sin^{2}{\left(x \right)} = 0$$
transform
$$- \left(2 \cos{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} = 0$$
$$2 \cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$2 w^{3} + w^{2} - 2 w - 1 = 0$$
transform
$$\left(- 2 w + \left(\left(w^{2} + \left(2 w^{3} - 2\right)\right) - 1\right)\right) + 2 = 0$$
or
$$\left(- 2 w + \left(\left(w^{2} + \left(2 w^{3} - 2 \cdot 1^{3}\right)\right) - 1^{2}\right)\right) + 2 = 0$$
$$- 2 \left(w - 1\right) + \left(\left(w^{2} - 1^{2}\right) + 2 \left(w^{3} - 1^{3}\right)\right) = 0$$
$$- 2 \left(w - 1\right) + \left(\left(w - 1\right) \left(w + 1\right) + 2 \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Take common factor -1 + w from the equation
we get:
$$\left(w - 1\right) \left(\left(\left(w + 1\right) + 2 \left(\left(w^{2} + w\right) + 1^{2}\right)\right) - 2\right) = 0$$
or
$$\left(w - 1\right) \left(2 w^{2} + 3 w + 1\right) = 0$$
then:
$$w_{1} = 1$$
and also
we get the equation
$$2 w^{2} + 3 w + 1 = 0$$
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
$$a = 2$$
$$b = 3$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{2} = - \frac{1}{2}$$
$$w_{3} = -1$$
The final answer for -1 + cos(x)^2 - 2*cos(x) + 2*cos(x)^3 = 0:
$$w_{1} = 1$$
$$w_{2} = - \frac{1}{2}$$
$$w_{3} = -1$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{2 \pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{3} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{3} = \pi n + \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
$$x_{5} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{5} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{5} = \pi n - \frac{\pi}{3}$$
$$x_{6} = \pi n + \operatorname{acos}{\left(w_{3} \right)} - \pi$$
$$x_{6} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{6} = \pi n$$
$$\left(2 \cos^{3}{\left(x \right)} - 2 \cos{\left(x \right)}\right) - \sin^{2}{\left(x \right)} = 0$$
transform
$$- \left(2 \cos{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} = 0$$
$$2 \cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$2 w^{3} + w^{2} - 2 w - 1 = 0$$
transform
$$\left(- 2 w + \left(\left(w^{2} + \left(2 w^{3} - 2\right)\right) - 1\right)\right) + 2 = 0$$
or
$$\left(- 2 w + \left(\left(w^{2} + \left(2 w^{3} - 2 \cdot 1^{3}\right)\right) - 1^{2}\right)\right) + 2 = 0$$
$$- 2 \left(w - 1\right) + \left(\left(w^{2} - 1^{2}\right) + 2 \left(w^{3} - 1^{3}\right)\right) = 0$$
$$- 2 \left(w - 1\right) + \left(\left(w - 1\right) \left(w + 1\right) + 2 \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Take common factor -1 + w from the equation
we get:
$$\left(w - 1\right) \left(\left(\left(w + 1\right) + 2 \left(\left(w^{2} + w\right) + 1^{2}\right)\right) - 2\right) = 0$$
or
$$\left(w - 1\right) \left(2 w^{2} + 3 w + 1\right) = 0$$
then:
$$w_{1} = 1$$
and also
we get the equation
$$2 w^{2} + 3 w + 1 = 0$$
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
$$a = 2$$
$$b = 3$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (2) * (1) = 1
Because D > 0, then the equation has two roots.
w2 = (-b + sqrt(D)) / (2*a)
w3 = (-b - sqrt(D)) / (2*a)
or
$$w_{2} = - \frac{1}{2}$$
$$w_{3} = -1$$
The final answer for -1 + cos(x)^2 - 2*cos(x) + 2*cos(x)^3 = 0:
$$w_{1} = 1$$
$$w_{2} = - \frac{1}{2}$$
$$w_{3} = -1$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{2 \pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{3} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{3} = \pi n + \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
$$x_{5} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{5} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{5} = \pi n - \frac{\pi}{3}$$
$$x_{6} = \pi n + \operatorname{acos}{\left(w_{3} \right)} - \pi$$
$$x_{6} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{6} = \pi n$$
Sum and product of roots
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sum
2*pi 2*pi - ---- + ---- 3 3
$$- \frac{2 \pi}{3} + \frac{2 \pi}{3}$$
=
0
$$0$$
product
-2*pi 2*pi
0*-----*----
3 3
$$\frac{2 \pi}{3} \cdot 0 \left(- \frac{2 \pi}{3}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
-2*pi
x2 = -----
3
$$x_{2} = - \frac{2 \pi}{3}$$
2*pi
x3 = ----
3
$$x_{3} = \frac{2 \pi}{3}$$
x3 = 2*pi/3
Numerical answer
[src]
x1 = -35.6047167406843
x2 = 98.4365698124802
x3 = -31.4159266901138
x4 = -6.28318515216187
x5 = -59.6902605042705
x6 = -81.6814090377032
x7 = 81.6814091574308
x8 = 96.342174710087
x9 = -48.1710873550435
x10 = 15.7079633858781
x11 = -41.8879020478639
x12 = 2.0943951023932
x13 = -85.870199198121
x14 = -53.4070752131038
x15 = -4.18879020478639
x16 = 4.18879020478639
x17 = -50.2654823076209
x18 = -100.530964743393
x19 = -65.9734457653993
x20 = -59.690259787532
x21 = -64.9262481741891
x22 = 28.2743342070522
x23 = 43.9822971693758
x24 = -12.5663700950596
x25 = -9.42477806893922
x26 = 52.3598775598299
x27 = -21.9911485914265
x28 = -8.37758040957278
x29 = -92.1533845053006
x30 = 56.5486676202915
x31 = -69.1150384949903
x32 = -28.2743337630989
x33 = 37.6991120028585
x34 = -52.3598775598299
x35 = 91.1061862679054
x36 = 87.9645943355569
x37 = 10526.4297846282
x38 = 72.2566312709154
x39 = 94.2477796093526
x40 = 62.8318529195469
x41 = -60.7374579694027
x42 = 18.8495557785441
x43 = -25.1327413598517
x44 = 8.37758040957278
x45 = 72.2566310277249
x46 = 48.1710873550435
x47 = -94.2477794635123
x48 = 92.1533845053006
x49 = 12.5663704648816
x50 = 0.0
x51 = -97.3893723535023
x52 = -37.6991118769989
x53 = -96.342174710087
x54 = -12.5663704539302
x55 = 56.5486674642584
x56 = -83.7758040957278
x57 = -90.0589894029074
x58 = -39.7935069454707
x59 = 65.9734454087465
x60 = 58.6430628670095
x61 = 46.0766922526503
x62 = -15.7079632962144
x63 = 79.5870138909414
x64 = -21.9911485864933
x65 = -46.0766922526503
x66 = 3.1415928179198
x67 = -79.5870138909414
x68 = -78.5398159768688
x69 = 31.4159266883903
x70 = -29.3215314335047
x71 = 65.9734457525392
x72 = -87.9645943589866
x73 = 37.6991120355511
x74 = 28.2743338652937
x75 = 34.5575190779589
x76 = -56.5486675979512
x77 = -72.2566309134271
x78 = -59.6902604569433
x79 = -43.9822971746331
x80 = 41.8879020478639
x81 = 75.3982238286389
x82 = -65.9734455091344
x83 = 21.9911485851557
x84 = 21.9911483137036
x85 = -75.3982238446723
x86 = 85.870199198121
x87 = 59.6902605332418
x88 = 50.2654824463584
x89 = 100.530964776097
x90 = 90.0589894029074
x91 = -73.3038285837618
x92 = 35.6047167406843
x93 = -2.0943951023932
x94 = 14.6607657167524
x95 = 6.28318528430958
x96 = 78.539816226929
x97 = -15.7079633869467
x97 = -15.7079633869467