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cos(x/2)=(1/2)

cos(x/2)=(1/2) equation

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Numerical solution:

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The solution

You have entered [src] 
   /x\      
cos|-| = 1/2
   \2/
cos⁡(x2)=12\cos{\left(\frac{x}{2} \right)} = \frac{1}{2}
Simplify
Detail solution
Given the equation
cos⁡(x2)=12\cos{\left(\frac{x}{2} \right)} = \frac{1}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x2=πn+acos⁡(12)\frac{x}{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}
x2=πn−π+acos⁡(12)\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}
Or
x2=πn+π3\frac{x}{2} = \pi n + \frac{\pi}{3}
x2=πn−2π3\frac{x}{2} = \pi n - \frac{2 \pi}{3}
, where n - is a integer
Divide both parts of the equation by
12\frac{1}{2}
we get the answer:
x1=2πn+2π3x_{1} = 2 \pi n + \frac{2 \pi}{3}
x2=2πn−4π3x_{2} = 2 \pi n - \frac{4 \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}
Simplify 
     10*pi
x2 = -----
       3
x2=10π3x_{2} = \frac{10 \pi}{3}
Simplify
Sum and product of roots [src] 
sum
    2*pi   10*pi
0 + ---- + -----
     3       3
(0+2π3)+10π3\left(0 + \frac{2 \pi}{3}\right) + \frac{10 \pi}{3} 
=
4*pi
4π4 \pi 
product
  2*pi 10*pi
1*----*-----
   3     3
10π3⋅1⋅2π3\frac{10 \pi}{3} \cdot 1 \cdot \frac{2 \pi}{3} 
=
     2
20*pi 
------
  9
20π29\frac{20 \pi^{2}}{9} 
20*pi^2/9
Numerical answer [src] 
x1 = -85.870199198121
x2 = -14.6607657167524
x3 = -39.7935069454707
x4 = 2.0943951023932
x5 = 73.3038285837618
x6 = -111.002940426839
x7 = -48.1710873550435
x8 = -2.0943951023932
x9 = 85.870199198121
x10 = 23.0383461263252
x11 = 64.9262481741891
x12 = -90.0589894029074
x13 = -429.350995990605
x14 = 52.3598775598299
x15 = 39.7935069454707
x16 = 60.7374579694027
x17 = 48.1710873550435
x18 = 98.4365698124802
x19 = -23.0383461263252
x20 = -27.2271363311115
x21 = -77.4926187885482
x22 = -64.9262481741891
x23 = -60.7374579694027
x24 = 90.0589894029074
x25 = 77.4926187885482
x26 = -278.554548618295
x27 = -437.728576400178
x28 = -98.4365698124802
x29 = 10.471975511966
x30 = -73.3038285837618
x31 = -10.471975511966
x32 = -265014.284281323
x33 = -35.6047167406843
x34 = 35.6047167406843
x35 = 2088.11191708602
x36 = -605.280184591634
x37 = 27.2271363311115
x38 = 14.6607657167524
x39 = -52.3598775598299
x39 = -52.3598775598299
The graph
cos(x/2)=(1/2) equation
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