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Sin(pi(x-3)/4)=(√2)/2 equation

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Numerical solution:

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The solution

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                    ___
   /pi*(x - 3)\   \/ 2 
sin|----------| = -----
   \    4     /     2
sin⁡(π(x−3)4)=22\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{\sqrt{2}}{2}
Simplify
Detail solution
Given the equation
sin⁡(π(x−3)4)=22\sin{\left(\frac{\pi \left(x - 3\right)}{4} \right)} = \frac{\sqrt{2}}{2}
- this is the simplest trigonometric equation
Divide both parts of the equation by -1

The equation is transformed to
sin⁡(πx4+π4)=−22\sin{\left(\frac{\pi x}{4} + \frac{\pi}{4} \right)} = - \frac{\sqrt{2}}{2}
This equation is transformed to
πx4+π4=2πn+asin⁡(−22)\frac{\pi x}{4} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}
πx4+π4=2πn−asin⁡(−22)+π\frac{\pi x}{4} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi
Or
πx4+π4=2πn−π4\frac{\pi x}{4} + \frac{\pi}{4} = 2 \pi n - \frac{\pi}{4}
πx4+π4=2πn+5π4\frac{\pi x}{4} + \frac{\pi}{4} = 2 \pi n + \frac{5 \pi}{4}
, where n - is a integer
Move
π4\frac{\pi}{4}
to right part of the equation
with the opposite sign, in total:
πx4=2πn−π2\frac{\pi x}{4} = 2 \pi n - \frac{\pi}{2}
πx4=2πn+π\frac{\pi x}{4} = 2 \pi n + \pi
Divide both parts of the equation by
π4\frac{\pi}{4}
we get the answer:
x1=4(2πn−π2)πx_{1} = \frac{4 \left(2 \pi n - \frac{\pi}{2}\right)}{\pi}
x2=4(2πn+π)πx_{2} = \frac{4 \left(2 \pi n + \pi\right)}{\pi}
The graph
0-80-60-40-2020406080-1001002-2
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -2
x1=−2x_{1} = -2 
x2 = 4
x2=4x_{2} = 4 
x2 = 4
Sum and product of roots [src] 
sum
-2 + 4
−2+4-2 + 4 
=
2
22 
product
-2*4
−8- 8 
=
-8
−8-8 
-8
Numerical answer [src] 
x1 = 94.0
x2 = 6.0
x3 = -74.0
x4 = -66.0
x5 = -36.0
x6 = 100.0
x7 = 60.0
x8 = -28.0
x9 = -52.0
x10 = -26.0
x11 = -2.0
x12 = -50.0
x13 = 68.0
x14 = 36.0
x15 = 46.0
x16 = 12.0
x17 = 28.0
x18 = 4.0
x19 = -90.0
x20 = 62.0
x21 = 70.0
x22 = -100.0
x23 = -92.0
x24 = -44.0
x25 = 52.0
x26 = -82.0
x27 = 54.0
x28 = 44.0
x29 = -76.0
x30 = 84.0
x31 = -20.0
x32 = -10.0
x33 = 86.0
x34 = 78.0
x35 = -98.0
x36 = -34.0
x37 = 76.0
x38 = -4.0
x39 = -42.0
x40 = 20.0
x41 = 22.0
x42 = -12.0
x43 = -18.0
x44 = -58.0
x45 = -68.0
x46 = 38.0
x47 = 30.0
x48 = 92.0
x49 = 14.0
x50 = -84.0
x51 = -60.0
x51 = -60.0
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