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cos(5x)=√2/2 equation

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             ___
           \/ 2 
cos(5*x) = -----
             2

\cos{\left(5 x \right)} = \frac{\sqrt{2}}{2}

Simplify

Detail solution

Given the equation

\cos{\left(5 x \right)} = \frac{\sqrt{2}}{2}

- this is the simplest trigonometric equation
This equation is transformed to

5 x = 2 \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}

5 x = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}

5 x = 2 \pi n + \frac{\pi}{4}

5 x = 2 \pi n - \frac{3 \pi}{4}

, where n - is a integer
Divide both parts of the equation by

5

we get the answer:

x_{1} = \frac{2 \pi n}{5} + \frac{\pi}{20}

x_{2} = \frac{2 \pi n}{5} - \frac{3 \pi}{20}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

      pi
x_1 = --
      20

x_{1} = \frac{\pi}{20}

Simplify

      7*pi
x_2 = ----
       20

x_{2} = \frac{7 \pi}{20}

Simplify

Sum and product of roots [src]

sum

pi   7*pi
-- + ----
20    20

\left(\frac{\pi}{20}\right) + \left(\frac{7 \pi}{20}\right)

2*pi
----
 5

\frac{2 \pi}{5}

Simplify

product

pi   7*pi
-- * ----
20    20

\left(\frac{\pi}{20}\right) * \left(\frac{7 \pi}{20}\right)

    2
7*pi 
-----
 400

\frac{7 \pi^{2}}{400}

Simplify

Numerical answer [src]

x1 = -85.6083998103219

x2 = 6.1261056745001

x3 = 73.9845069920396

x4 = 36.2853951489621

x5 = -84.037603483527

x6 = -51.6791991515521

x7 = 66.4446846234241

x8 = -43.8252175175776

x9 = 34.0862802914493

x10 = -42.8827397215007

x11 = 46.3384916404494

x12 = 10.2101761241668

x13 = 30.0022098417825

x14 = -97.8606111593221

x15 = -60.1614993162445

x16 = 64.2455697659113

x17 = 68.015480950219

x18 = 2.35619449019234

x19 = 11.1526539202438

x20 = 98.174770424681

x21 = -31.5730061685774

x22 = 16.1792021659874

x23 = -525.117212047534

x24 = 202.475646523862

x25 = 70.2145958077319

x26 = 96.9181333632451

x27 = -3.92699081698724

x28 = -61.7322956430394

x29 = -77.7544181763474

x30 = 0.15707963267949

x31 = 37.856191475757

x32 = 60.1614993162445

x33 = -81.8384886260141

x34 = -27.8030949842697

x35 = 89.3783109946296

x36 = -80.5818515645782

x37 = 26.2322986574748

x38 = -21.5199096770901

x39 = 40.0553063332699

x40 = 22.462387473167

x41 = 32.8296432300133

x42 = 24.0331837999619

x43 = 13.9800873084746

x44 = 61.4181363776805

x45 = -17.7499984927823

x46 = -7.69690200129499

x47 = 27.8030949842697

x48 = -70.2145958077319

x49 = 90.3207887907066

x50 = -65.1880475619882

x51 = -68.015480950219

x52 = -47.9092879672443

x53 = 100.373885282194

x54 = 88.1216739331937

x55 = 80.2676922992192

x56 = 47.9092879672443

x57 = -87.8075146678347

x58 = 71.7853921345268

x59 = 56.3915881319368

x60 = -79.0110552377833

x61 = -41.6261026600648

x62 = -71.7853921345268

x63 = -75.5553033188345

x64 = -73.9845069920396

x65 = 3.92699081698724

x66 = 17.4358392274234

x67 = 78.0685774417064

x68 = -95.6614963018092

x69 = 76.8119403802704

x70 = -13.9800873084746

x71 = -15.2367243699105

x72 = -50.1084028247572

x73 = 94.0906999750143

x74 = 81.8384886260141

x75 = -9.89601685880785

x76 = -53.878314009065

x77 = 84.037603483527

x78 = -19.9491133502952

x79 = 20.2632726156542

x80 = -63.9314105005523

x81 = 57.9623844587317

x82 = 91.8915851175014

x83 = -30.0022098417825

x84 = -16.1792021659874

x85 = -91.8915851175014

x86 = -94.0906999750143

x87 = -57.9623844587317

x88 = -37.856191475757

x89 = 12.4092909816797

x90 = -40.0553063332699

x91 = 50.1084028247572

x92 = -33.7721210260903

x93 = -6.1261056745001

x94 = 54.1924732744239

x95 = 44.1393767829366

x95 = 44.1393767829366

The graph