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Similar expressions
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2sin²пx+5sinпx-7=0 equation
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The solution
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[src]
2 2*sin (pi*x) + 5*sin(pi*x) - 7 = 0
$$\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0$$
transform
$$2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)} - 7 = 0$$
$$\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0$$
Do replacement
$$w = \sin{\left(\pi x \right)}$$
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
$$a = 2$$
$$b = 5$$
$$c = -7$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 1$$
$$w_{2} = - \frac{7}{2}$$
do backward replacement
$$\sin{\left(\pi x \right)} = w$$
Given the equation
$$\sin{\left(\pi x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\pi$$
substitute w:
$$x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(w_{1} \right)}}{\pi}$$
$$x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(1 \right)}}{\pi}$$
$$x_{1} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}$$
$$x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(w_{2} \right)}}{\pi}$$
$$x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}$$
$$x_{2} = \frac{2 \pi n - \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}$$
$$x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi}{\pi}$$
$$x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi}{\pi}$$
$$x_{3} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}$$
$$x_{4} = \frac{2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi}{\pi}$$
$$x_{4} = \frac{2 \pi n + \pi - \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}$$
$$x_{4} = \frac{2 \pi n + \pi + \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}$$
$$\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0$$
transform
$$2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)} - 7 = 0$$
$$\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0$$
Do replacement
$$w = \sin{\left(\pi x \right)}$$
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
$$a = 2$$
$$b = 5$$
$$c = -7$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (2) * (-7) = 81
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 1$$
$$w_{2} = - \frac{7}{2}$$
do backward replacement
$$\sin{\left(\pi x \right)} = w$$
Given the equation
$$\sin{\left(\pi x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\pi$$
substitute w:
$$x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(w_{1} \right)}}{\pi}$$
$$x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(1 \right)}}{\pi}$$
$$x_{1} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}$$
$$x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(w_{2} \right)}}{\pi}$$
$$x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}$$
$$x_{2} = \frac{2 \pi n - \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}$$
$$x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi}{\pi}$$
$$x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi}{\pi}$$
$$x_{3} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}$$
$$x_{4} = \frac{2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi}{\pi}$$
$$x_{4} = \frac{2 \pi n + \pi - \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}$$
$$x_{4} = \frac{2 \pi n + \pi + \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}$$
Rapid solution
[src]
x1 = 1/2
$$x_{1} = \frac{1}{2}$$
pi + re(asin(7/2)) I*im(asin(7/2))
x2 = ------------------ + ---------------
pi pi
$$x_{2} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}$$
re(asin(7/2)) I*im(asin(7/2))
x3 = - ------------- - ---------------
pi pi
$$x_{3} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}$$
x3 = -re(asin(7/2))/pi - i*im(asin(7/2))/pi
Sum and product of roots
[src]
sum
1 pi + re(asin(7/2)) I*im(asin(7/2)) re(asin(7/2)) I*im(asin(7/2)) - + ------------------ + --------------- + - ------------- - --------------- 2 pi pi pi pi
$$\left(\frac{1}{2} + \left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)$$
=
1 pi + re(asin(7/2)) re(asin(7/2)) - + ------------------ - ------------- 2 pi pi
$$- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} + \frac{1}{2} + \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi}$$
product
pi + re(asin(7/2)) I*im(asin(7/2))
------------------ + ---------------
pi pi / re(asin(7/2)) I*im(asin(7/2))\
------------------------------------*|- ------------- - ---------------|
2 \ pi pi /
$$\frac{\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}}{2} \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)$$
=
-(I*im(asin(7/2)) + re(asin(7/2)))*(pi + I*im(asin(7/2)) + re(asin(7/2)))
--------------------------------------------------------------------------
2
2*pi
$$- \frac{\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right)}{2 \pi^{2}}$$
-(i*im(asin(7/2)) + re(asin(7/2)))*(pi + i*im(asin(7/2)) + re(asin(7/2)))/(2*pi^2)
Numerical answer
[src]
x1 = -37.5000001005768
x2 = -61.5000001005768
x3 = 86.4999998994232
x4 = -55.5000001005768
x5 = 6.49999989942322
x6 = -29.5000001005768
x7 = -83.5000001005768
x8 = 34.4999998994232
x9 = 40.4999998994232
x10 = -69.5000001005768
x11 = 52.4999998994232
x12 = -11.5000001005768
x13 = 46.4999998994232
x14 = -67.5000001005768
x15 = 88.4999998994232
x16 = 70.4999998994232
x17 = 28.4999998994232
x18 = 56.4999998994232
x19 = -49.5000001005768
x20 = -97.5000001005768
x21 = -25.5000001005768
x22 = 30.4999998994232
x23 = -35.5000001005768
x24 = -1.50000010057678
x25 = -21.5000001005768
x26 = -73.5000001005768
x27 = -79.5000001005768
x28 = 92.4999998994232
x29 = 24.4999998994232
x30 = -15.5000001005768
x31 = 16.4999998994232
x32 = 80.4999998994232
x33 = 84.4999998994232
x34 = 38.4999998994232
x35 = -63.5000001005768
x36 = 14.4999998994232
x37 = 60.4999998994232
x38 = -87.5000001005768
x39 = 68.4999998994232
x40 = -91.5000001005768
x41 = 42.4999998994232
x42 = -27.5000001005768
x43 = 36.4999998994232
x44 = -85.5000001005768
x45 = -65.5000001005768
x46 = -81.5000001005768
x47 = -53.5000001005768
x48 = -9.50000010057678
x49 = 4.49999989942322
x50 = -99.5000001005768
x51 = 66.4999998994232
x52 = 18.4999998994232
x53 = -59.5000001005768
x54 = 96.4999998994232
x55 = 0.499999899423223
x56 = 32.4999998994232
x57 = 78.4999998994232
x58 = -71.5000001005768
x59 = -5.50000010057678
x60 = 8.49999989942322
x61 = -19.5000001005768
x62 = -89.5000001005768
x63 = 48.4999998994232
x64 = -75.5000001005768
x65 = -13.5000001005768
x66 = -45.5000001005768
x67 = -77.5000001005768
x68 = 20.4999998994232
x69 = 10.4999998994232
x70 = -23.5000001005768
x71 = 74.4999998994232
x72 = 72.4999998994232
x73 = -57.5000001005768
x74 = 22.4999998994232
x75 = 94.4999998994232
x76 = -3.50000010057678
x77 = -51.5000001005768
x78 = -33.5000001005768
x79 = 64.4999998994232
x80 = 100.499999899423
x81 = -41.5000001005768
x82 = -93.5000001005768
x83 = -7.50000010057678
x84 = 26.4999998994232
x85 = 2.49999989942322
x86 = 50.4999998994232
x87 = 54.4999998994232
x88 = 12.4999998994232
x89 = -47.5000001005768
x90 = 76.4999998994232
x91 = 44.4999998994232
x92 = 58.4999998994232
x93 = -95.5000001005768
x94 = -17.5000001005768
x95 = 98.4999998994232
x96 = -43.5000001005768
x97 = 82.4999998994232
x98 = 62.4999998994232
x99 = 90.4999998994232
x100 = -31.5000001005768
x101 = -39.5000001005768
x101 = -39.5000001005768