Mister Exam

# sin(8x-(pi/3))=0 equation

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#### Numerical solution:

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### The solution

You have entered [src]
   /      pi\
sin|8*x - --| = 0
\      3 /    
$$\sin{\left(8 x - \frac{\pi}{3} \right)} = 0$$
Detail solution
Given the equation
$$\sin{\left(8 x - \frac{\pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0

We get:
$$\sin{\left(8 x - \frac{\pi}{3} \right)} = 0$$
Divide both parts of the equation by -1

The equation is transformed to
$$\cos{\left(8 x + \frac{\pi}{6} \right)} = 0$$
This equation is transformed to
$$8 x + \frac{\pi}{6} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$8 x + \frac{\pi}{6} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$8 x + \frac{\pi}{6} = \pi n + \frac{\pi}{2}$$
$$8 x + \frac{\pi}{6} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$8 x = \pi n + \frac{\pi}{3}$$
$$8 x = \pi n - \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$8$$
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{24}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{12}$$
The graph
Sum and product of roots [src]
sum
pi   pi
-- + --
24   6 
$$\frac{\pi}{24} + \frac{\pi}{6}$$
=
5*pi
----
24 
$$\frac{5 \pi}{24}$$
product
pi pi
--*--
24 6 
$$\frac{\pi}{24} \frac{\pi}{6}$$
=
  2
pi
---
144
$$\frac{\pi^{2}}{144}$$
pi^2/144
Rapid solution [src]
     pi
x1 = --
24
$$x_{1} = \frac{\pi}{24}$$
     pi
x2 = --
6 
$$x_{2} = \frac{\pi}{6}$$
x2 = pi/6
x1 = 90.0589894029074
x2 = 95.9494756283883
x3 = 28.012534494509
x4 = 40.1862060271694
x5 = -15.9697626557481
x6 = 0.130899693899575
x7 = 14.2680666350536
x8 = -73.3038285837618
x9 = -34.0339204138894
x10 = -97.6511716490827
x11 = -9.29387826686981
x12 = -79.9797129726402
x13 = -85.870199198121
x14 = 58.2503637853107
x15 = -83.9067037896274
x16 = -35.997415822383
x17 = -98.0438707307815
x18 = 80.2415123604393
x19 = -27.7507351067098
x20 = 6.02138591938044
x21 = 88.0954939944138
x22 = 22.1220482690281
x23 = 36.2592152101822
x24 = 66.1043454192852
x25 = 46.0766922526503
x26 = 86.1319985859202
x27 = 44.1131968441567
x28 = 73.9583270532597
x29 = 48.0401876611439
x30 = -89.7971900151083
x31 = -25.7872396982162
x32 = -91.7606854236019
x33 = 100.269165527074
x34 = -69.7695368484733
x35 = 38.2227106186758
x36 = -59.9520598060052
x37 = -87.8336946066146
x38 = 51.9671784781312
x39 = 7.98488132787406
x40 = -71.733032256967
x41 = -56.025068989018
x42 = 70.0313362362725
x43 = -78.0162175641465
x44 = -39.9244066393703
x45 = 82.2050077689329
x46 = -41.8879020478639
x47 = -61.9155552144988
x48 = -511.556003759538
x49 = -100.007366139275
x50 = -19.8967534727354
x51 = -67.8060414399797
x52 = -49.7418836818384
x53 = 46.8620904160477
x54 = 92.022484811401
x55 = 16.2315620435473
x56 = 62.177354602298
x57 = -3.79609112308767
x58 = -51.705379090332
x59 = -65.8425460314861
x60 = 60.2138591938044
x61 = 78.2780169519457
x62 = -54.4542726622231
x63 = -47.7783882733448
x64 = -7.72308194007491
x65 = 29.9760299030026
x66 = -45.8148928648512
x67 = -217.424391567194
x68 = -14.0062672472545
x69 = -12.0427718387609
x70 = -43.8513974563575
x71 = -1.83259571459405
x72 = 56.2868683768171
x73 = -29.7142305152035
x74 = -17.9332580642417
x75 = 34.2957198016886
x76 = -93.7241808320955
x77 = 26.0490390860154
x78 = -57.9885643975116
x79 = -37.9609112308767
x80 = 93.9859802198946
x81 = 84.1685031774265
x82 = 50.0036830696375
x83 = 42.1497014356631
x84 = -23.8237442897226
x85 = 18.1950574520409
x86 = -21.860248881229
x87 = -5.75958653158129
x88 = 24.0855436775217
x89 = 64.1408500107916
x90 = 4.05789051088682
x91 = 20.1585528605345
x92 = 71.9948316447661
x93 = -76.4454212373516
x94 = -63.8790506229925
x95 = 2.0943951023932
x96 = -81.9432083811338
x97 = 68.0678408277789
x97 = 68.0678408277789