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You entered:

sin(6*x-pi/6)=1/2

sin((6*x - pi)/6) = 1/2

sin(6*(x - pi)/6) = 1/2

sin(6*x - pi/6) = 1/2

sin(6*x - pi/6) = 1/2

sin(6*x-pi/6)=1/2 equation

The teacher will be very surprised to see your correct solution 😉

You have entered [src]

   /      pi\      
sin|6*x - --| = 1/2
   \      6 /

\sin{\left(6 x - \frac{\pi}{6} \right)} = \frac{1}{2}

Simplify

Detail solution

Given the equation

\sin{\left(6 x - \frac{\pi}{6} \right)} = \frac{1}{2}

- this is the simplest trigonometric equation
Divide both parts of the equation by

-1

The equation is transformed to

\cos{\left(6 x + \frac{\pi}{3} \right)} = - \frac{1}{2}

This equation is transformed to

6 x + \frac{\pi}{3} = 2 \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}

6 x + \frac{\pi}{3} = 2 \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}

6 x + \frac{\pi}{3} = 2 \pi n + \frac{2 \pi}{3}

6 x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{3}

, where n - is a integer
Move

\frac{\pi}{3}

to right part of the equation with the opposite sign, in total:

6 x = 2 \pi n + \frac{\pi}{3}

6 x = 2 \pi n - \frac{2 \pi}{3}

Divide both parts of the equation by

6

we get the answer:

x_{1} = \frac{\pi n}{3} + \frac{\pi}{18}

x_{2} = \frac{\pi n}{3} - \frac{\pi}{9}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

      pi
x_1 = --
      18

x_{1} = \frac{\pi}{18}

Simplify

      pi
x_2 = --
      6

x_{2} = \frac{\pi}{6}

Simplify

Sum and product of roots [src]

sum

pi   pi
-- + --
18   6

\left(\frac{\pi}{18}\right) + \left(\frac{\pi}{6}\right)

2*pi
----
 9

\frac{2 \pi}{9}

Simplify

product

pi   pi
-- * --
18   6

\left(\frac{\pi}{18}\right) * \left(\frac{\pi}{6}\right)

  2
pi 
---
108

\frac{\pi^{2}}{108}

Simplify

Numerical answer [src]

x1 = 39.9680398706701

x2 = -91.9788515801012

x3 = -71.733032256967

x4 = 68.2423737529783

x5 = 38.2227106186758

x6 = 4.36332312998582

x7 = 92.3279174305

x8 = -26.0054058547155

x9 = 56.025068989018

x10 = -23.9110107523223

x11 = 53.9306738866248

x12 = 29.4960643587042

x13 = -19.7222205475359

x14 = -39.6189740202713

x15 = -4.01425727958696

x16 = -85.6956662729216

x17 = 60.2138591938044

x18 = 2.26892802759263

x19 = -5.75958653158129

x20 = -43.8077642250577

x21 = -29.845130209103

x22 = 80.1106126665397

x23 = -53.9306738866248

x24 = -73.8274273593601

x25 = 46.2512251778497

x26 = 17.9768912955416

x27 = 94.7713783832921

x28 = -31.9395253114962

x29 = 51.4872129338327

x30 = -7.85398163397448

x31 = 47.2984227290463

x32 = 86.0447321233204

x33 = 21.4675497995303

x34 = -100.007366139275

x35 = -45.9021593274509

x36 = 48.3456202802429

x37 = 73.4783615089613

x38 = 12.0427718387609

x39 = -14.1371669411541

x40 = 16.2315620435473

x41 = -63.704517697793

x42 = -41.7133691226645

x43 = 24.2600766027212

x44 = -61.6101225953998

x45 = -49.7418836818384

x46 = 14.1371669411541

x47 = 44.1568300754565

x48 = 0.174532925199433

x49 = 100.007366139275

x50 = 7.85398163397448

x51 = 66.1479786505851

x52 = -36.1283155162826

x53 = 75.9218224617533

x54 = 20.0712863979348

x55 = 78.0162175641465

x56 = -95.8185759344887

x57 = -87.7900613753148

x58 = -93.7241808320955

x59 = 22.165681500328

x60 = -34.0339204138894

x61 = 97.9129710368819

x62 = 36.1283155162826

x63 = -51.8362787842316

x64 = -78.0162175641465

x65 = -12.0427718387609

x66 = -80.1106126665397

x67 = -75.9218224617533

x68 = 34.0339204138894

x69 = -97.9129710368819

x70 = -21.8166156499291

x71 = -83.6012711705284

x72 = -56.025068989018

x73 = 90.2335223281068

x74 = 88.1391272257137

x75 = -65.7989128001862

x76 = -67.8933079025794

x77 = -47.9965544298441

x78 = -58.1194640914112

x79 = -27.7507351067098

x80 = 70.3367688553715

x81 = 42.0624349730633

x82 = -1.91986217719376

x83 = -89.884456477708

x84 = -9.94837673636768

x85 = 64.0535835481919

x86 = 61.261056745001

x87 = 82.2050077689329

x88 = -17.6278254451427

x89 = -69.9877030049726

x90 = 58.1194640914112

x91 = 31.9395253114962

x92 = 26.3544717051144

x93 = 9.94837673636768

x93 = 9.94837673636768

The graph