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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+2π3\frac{x}{2} = \pi n + \frac{2 \pi}{3}
x2=πn−π3\frac{x}{2} = \pi n - \frac{\pi}{3}
, where n - is a integer
Divide both parts of the equation by
12\frac{1}{2}
we get the answer:
x1=2πn+4π3x_{1} = 2 \pi n + \frac{4 \pi}{3}
x2=2πn−2π3x_{2} = 2 \pi n - \frac{2 \pi}{3}
The graph
0-80-60-40-2020406080-1001002-2
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
    4*pi   8*pi
0 + ---- + ----
     3      3
(0+4π3)+8π3\left(0 + \frac{4 \pi}{3}\right) + \frac{8 \pi}{3} 
=
4*pi
4π4 \pi 
product
  4*pi 8*pi
1*----*----
   3    3
8π3⋅1⋅4π3\frac{8 \pi}{3} \cdot 1 \cdot \frac{4 \pi}{3} 
=
     2
32*pi 
------
  9
32π29\frac{32 \pi^{2}}{9} 
32*pi^2/9
Rapid solution [src] 
     4*pi
x1 = ----
      3
x1=4π3x_{1} = \frac{4 \pi}{3}
Simplify 
     8*pi
x2 = ----
      3
x2=8π3x_{2} = \frac{8 \pi}{3}
Simplify
Numerical answer [src] 
x1 = 54.4542726622231
x2 = -192.684349420174
x3 = 20.943951023932
x4 = 58.6430628670095
x5 = -67.0206432765823
x6 = -71.2094334813686
x7 = -58.6430628670095
x8 = -79.5870138909414
x9 = -83.7758040957278
x10 = -20.943951023932
x11 = 96.342174710087
x12 = 67.0206432765823
x13 = 71.2094334813686
x14 = -92.1533845053006
x15 = -29.3215314335047
x16 = 4.18879020478639
x17 = 8.37758040957278
x18 = -117.286125734019
x19 = 33.5103216382911
x20 = 83.7758040957278
x21 = -96.342174710087
x22 = 16.7551608191456
x23 = 41.8879020478639
x24 = -41.8879020478639
x25 = -33.5103216382911
x26 = 29.3215314335047
x27 = -8.37758040957278
x28 = -4.18879020478639
x29 = -46.0766922526503
x30 = -16.7551608191456
x31 = -54.4542726622231
x32 = 79.5870138909414
x33 = 46.0766922526503
x34 = 92.1533845053006
x34 = 92.1533845053006
The graph
cos(x/2)=-(1/2) equation
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