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sin(x/3)=-1/2

sin(x/3)=-1/2 equation

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Numerical solution:

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The solution

You have entered [src]
   /x\       
sin|-| = -1/2
   \3/       
$$\sin{\left(\frac{x}{3} \right)} = - \frac{1}{2}$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{3} \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{3} = 2 \pi n - \frac{\pi}{6}$$
$$\frac{x}{3} = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n - \frac{\pi}{2}$$
$$x_{2} = 6 \pi n + \frac{7 \pi}{2}$$
The graph
Rapid solution [src]
     -pi 
x1 = ----
      2  
$$x_{1} = - \frac{\pi}{2}$$
     7*pi
x2 = ----
      2  
$$x_{2} = \frac{7 \pi}{2}$$
x2 = 7*pi/2
Sum and product of roots [src]
sum
  pi   7*pi
- -- + ----
  2     2  
$$- \frac{\pi}{2} + \frac{7 \pi}{2}$$
=
3*pi
$$3 \pi$$
product
-pi  7*pi
----*----
 2    2  
$$- \frac{\pi}{2} \frac{7 \pi}{2}$$
=
     2
-7*pi 
------
  4   
$$- \frac{7 \pi^{2}}{4}$$
-7*pi^2/4
Numerical answer [src]
x1 = -76.9690200129499
x2 = -359.712358836031
x3 = 86.3937979737193
x4 = -39.2699081698724
x5 = -58.1194640914112
x6 = -83.2522053201295
x7 = 92.6769832808989
x8 = -20.4203522483337
x9 = -102.101761241668
x10 = 67.5442420521806
x11 = -26.7035375555132
x12 = -64.4026493985908
x13 = -1.5707963267949
x14 = 17.2787595947439
x15 = -95.8185759344887
x16 = 10.9955742875643
x17 = 36.1283155162826
x18 = 48.6946861306418
x19 = 111.526539202438
x20 = -7.85398163397448
x21 = 73.8274273593601
x22 = 29.845130209103
x23 = -45.553093477052
x24 = 54.9778714378214
x24 = 54.9778714378214
The graph
sin(x/3)=-1/2 equation