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sin(x/4)=1/2

sin(x/4)=1/2 equation

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Numerical solution:

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The solution

You have entered [src] 
   /x\      
sin|-| = 1/2
   \4/
sin⁡(x4)=12\sin{\left(\frac{x}{4} \right)} = \frac{1}{2}
Simplify
Detail solution
Given the equation
sin⁡(x4)=12\sin{\left(\frac{x}{4} \right)} = \frac{1}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x4=2πn+asin⁡(12)\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x4=2πn−asin⁡(12)+π\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
Or
x4=2πn+π6\frac{x}{4} = 2 \pi n + \frac{\pi}{6}
x4=2πn+5π6\frac{x}{4} = 2 \pi n + \frac{5 \pi}{6}
, where n - is a integer
Divide both parts of the equation by
14\frac{1}{4}
we get the answer:
x1=8πn+2π3x_{1} = 8 \pi n + \frac{2 \pi}{3}
x2=8πn+10π3x_{2} = 8 \pi n + \frac{10 \pi}{3}
The graph
0-80-60-40-2020406080-1001002-2
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
     2*pi
x1 = ----
      3
x1=2π3x_{1} = \frac{2 \pi}{3} 
     10*pi
x2 = -----
       3
x2=10π3x_{2} = \frac{10 \pi}{3} 
x2 = 10*pi/3
Sum and product of roots [src] 
sum
2*pi   10*pi
---- + -----
 3       3
2π3+10π3\frac{2 \pi}{3} + \frac{10 \pi}{3} 
=
4*pi
4π4 \pi 
product
2*pi 10*pi
----*-----
 3     3
2π310π3\frac{2 \pi}{3} \frac{10 \pi}{3} 
=
     2
20*pi 
------
  9
20π29\frac{20 \pi^{2}}{9} 
20*pi^2/9
Numerical answer [src] 
x1 = 228.289066160858
x2 = 3570.9436495804
x3 = 2.0943951023932
x4 = -14.6607657167524
x5 = -23.0383461263252
x6 = 102.625360017267
x7 = 27.2271363311115
x8 = -90.0589894029074
x9 = 52.3598775598299
x10 = 35.6047167406843
x11 = 60.7374579694027
x12 = -73.3038285837618
x13 = 77.4926187885482
x14 = 85.870199198121
x15 = 10.471975511966
x16 = -98.4365698124802
x17 = -64.9262481741891
x18 = -48.1710873550435
x19 = -39.7935069454707
x19 = -39.7935069454707
The graph
sin(x/4)=1/2 equation
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