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Similar expressions
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(sin(2*pi*x))/(4x-1)=1/(4x-1) equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
sin(2*pi*x) 1 ----------- = ------- 4*x - 1 4*x - 1
$$\frac{\sin{\left(2 \pi x \right)}}{4 x - 1} = \frac{1}{4 x - 1}$$
Detail solution
Given the equation
$$\frac{\sin{\left(2 \pi x \right)}}{4 x - 1} = \frac{1}{4 x - 1}$$
transform
$$\frac{- 4 x + \sin{\left(2 \pi x \right)}}{4 x - 1} = 0$$
$$-1 + \frac{\sin{\left(2 \pi x \right)}}{4 x - 1} - \frac{1}{4 x - 1} = 0$$
Do replacement
$$w = \sin{\left(2 \pi x \right)}$$
Given the equation:
$$-1 + \frac{\sin{\left(2 \pi x \right)}}{4 x - 1} - \frac{1}{4 x - 1} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
so we get the equation
$$\frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x - 1$$
$$\frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x - 1$$
Expand brackets in the left part
Looking for similar summands in the left part:
Move free summands (without w)
from left part to right part, we given:
$$1 + \frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x$$
This equation has no roots
do backward replacement
$$\sin{\left(2 \pi x \right)} = w$$
Given the equation
$$\sin{\left(2 \pi x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 \pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 \pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 \pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 \pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2 \pi$$
substitute w:
$$\frac{\sin{\left(2 \pi x \right)}}{4 x - 1} = \frac{1}{4 x - 1}$$
transform
$$\frac{- 4 x + \sin{\left(2 \pi x \right)}}{4 x - 1} = 0$$
$$-1 + \frac{\sin{\left(2 \pi x \right)}}{4 x - 1} - \frac{1}{4 x - 1} = 0$$
Do replacement
$$w = \sin{\left(2 \pi x \right)}$$
Given the equation:
$$-1 + \frac{\sin{\left(2 \pi x \right)}}{4 x - 1} - \frac{1}{4 x - 1} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
a1 = sin(2*pi*x)
b1 = -1 + 4*x
a2 = 1
b2 = 1/(1 + 1/(-1 + 4*x))
so we get the equation
$$\frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x - 1$$
$$\frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x - 1$$
Expand brackets in the left part
sin2*pi*x1+1/-1+4*x) = -1 + 4*x
Looking for similar summands in the left part:
sin(2*pi*x)/(1 + 1/(-1 + 4*x)) = -1 + 4*x
Move free summands (without w)
from left part to right part, we given:
$$1 + \frac{\sin{\left(2 \pi x \right)}}{1 + \frac{1}{4 x - 1}} = 4 x$$
This equation has no roots
do backward replacement
$$\sin{\left(2 \pi x \right)} = w$$
Given the equation
$$\sin{\left(2 \pi x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 \pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 \pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 \pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 \pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2 \pi$$
substitute w:
Numerical answer
[src]
x1 = 18.25
x2 = 16.25
x3 = -41.75
x4 = -3.75
x5 = 92.25
x6 = -23.75
x7 = 6.25
x8 = 50.25
x9 = -59.75
x10 = 80.25
x11 = 24.25
x12 = 62.25
x13 = -15.75
x14 = -73.75
x15 = -69.75
x16 = 22.25
x17 = 96.25
x18 = 78.25
x19 = -9.75
x20 = 68.25
x21 = -49.75
x22 = 94.25
x23 = 88.25
x24 = 34.25
x25 = 42.25
x26 = 74.25
x27 = 76.25
x28 = -91.75
x29 = 64.25
x30 = 2.25
x31 = 4.25
x32 = -7.75
x33 = -71.75
x34 = -53.75
x35 = 38.25
x36 = -87.75
x37 = -35.75
x38 = 12.25
x39 = -77.75
x40 = -21.75
x41 = 30.25
x42 = -1.75
x43 = -57.75
x44 = -81.75
x45 = 72.25
x46 = 60.25
x47 = -5.75
x48 = -17.75
x49 = -25.75
x50 = -19.75
x51 = 14.25
x52 = 100.25
x53 = 90.25
x54 = -61.75
x55 = -43.75
x56 = 32.25
x57 = -63.75
x58 = -33.75
x59 = 10.25
x60 = -93.75
x61 = 58.25
x62 = 28.25
x63 = -67.75
x64 = -27.75
x65 = 86.25
x66 = -97.75
x67 = -75.75
x68 = 84.25
x69 = -83.75
x70 = -31.75
x71 = 66.25
x72 = 36.25
x73 = 70.25
x74 = -13.75
x75 = -45.75
x76 = -11.75
x77 = 40.25
x78 = -85.75
x79 = -47.75
x80 = -55.75
x81 = 20.25
x82 = 46.25
x83 = -39.75
x84 = -51.75
x85 = -79.75
x86 = -95.75
x87 = -29.75
x88 = 8.25
x89 = -89.75
x90 = -37.75
x91 = 98.25
x92 = 44.25
x93 = 56.25
x94 = 26.25
x95 = -99.75
x96 = 82.25
x97 = -65.75
x98 = 48.25
x99 = 54.25
x100 = 52.25
x100 = 52.25