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Equation:
Equation x/4+x=4
Equation x^2+11=0
Equation x+y=5
Equation 1*(x-1)*(x-3)=0
Express {x} in terms of y where:
-11*x-3*y=14
-1*x-17*y=-12
20*x+18*y=-9
-2*x+20*y=-2
Identical expressions
cos4xcos2xcosx= zero , one hundred and twenty-five
co sinus of e of 4x co sinus of e of 2x co sinus of e of x equally 0,125
co sinus of e of 4x co sinus of e of 2x co sinus of e of x equally zero , one hundred and twenty minus five
cos4xcos2xcosx=O,125

cos4xcos2xcosx=0,125 equation

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

The solution

You have entered [src]

cos(4*x)*cos(2*x)*cos(x) = 1/8

$$\cos{\left(2 x \right)} \cos{\left(4 x \right)} \cos{\left(x \right)} = \frac{1}{8}$$

Simplify

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

     -pi 
x1 = ----
      3

$$x_{1} = - \frac{\pi}{3}$$

     -2*pi
x2 = -----
       7

$$x_{2} = - \frac{2 \pi}{7}$$

     -pi 
x3 = ----
      9

$$x_{3} = - \frac{\pi}{9}$$

     pi
x4 = --
     9

$$x_{4} = \frac{\pi}{9}$$

     2*pi
x5 = ----
      7

$$x_{5} = \frac{2 \pi}{7}$$

     pi
x6 = --
     3

$$x_{6} = \frac{\pi}{3}$$

     7*pi
x7 = ----
      9

$$x_{7} = \frac{7 \pi}{9}$$

     6*pi
x8 = ----
      7

$$x_{8} = \frac{6 \pi}{7}$$

           / 7 ____\
x9 = -I*log\-\/ -1 /

$$x_{9} = - i \log{\left(- \sqrt[7]{-1} \right)}$$

            /     2/9\
x10 = -I*log\-(-1)   /

$$x_{10} = - i \log{\left(- \left(-1\right)^{\frac{2}{9}} \right)}$$

                                                   /   /pi\\
                 /    _____________________\       |cos|--||
                 |   /    2/pi\      2/pi\ |       |   \18/|
x11 = -pi - I*log|  /  cos |--| + sin |--| | + atan|-------|
                 \\/       \18/       \18/ /       |   /pi\|
                                                   |sin|--||
                                                   \   \18//

$$x_{11} = - \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)}$$

               /   /pi\\                                   
               |cos|--||        /    _____________________\
               |   \18/|        |   /    2/pi\      2/pi\ |
x12 = pi - atan|-------| - I*log|  /  cos |--| + sin |--| |
               |   /pi\|        \\/       \18/       \18/ /
               |sin|--||                                   
               \   \18//

$$x_{12} = - \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \pi$$

                                                   /   /pi\\
                 /    _____________________\       |cos|--||
                 |   /    2/pi\      2/pi\ |       |   \14/|
x13 = -pi - I*log|  /  cos |--| + sin |--| | + atan|-------|
                 \\/       \14/       \14/ /       |   /pi\|
                                                   |sin|--||
                                                   \   \14//

$$x_{13} = - \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)}$$

               /   /pi\\                                   
               |cos|--||        /    _____________________\
               |   \14/|        |   /    2/pi\      2/pi\ |
x14 = pi - atan|-------| - I*log|  /  cos |--| + sin |--| |
               |   /pi\|        \\/       \14/       \14/ /
               |sin|--||                                   
               \   \14//

$$x_{14} = - \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \pi$$

x14 = -atan(cos(pi/14)/sin(pi/14)) - i*log(sqrt(sin(pi/14)^2 + cos(pi/14)^2)) + pi

Sum and product of roots [src]

sum

                                                                                                                                  /   /pi\\            /   /pi\\                                                                                   /   /pi\\            /   /pi\\                                   
                                                                                                /    _____________________\       |cos|--||            |cos|--||        /    _____________________\              /    _____________________\       |cos|--||            |cos|--||        /    _____________________\
  pi   2*pi   pi   pi   2*pi   pi   7*pi   6*pi        / 7 ____\        /     2/9\              |   /    2/pi\      2/pi\ |       |   \18/|            |   \18/|        |   /    2/pi\      2/pi\ |              |   /    2/pi\      2/pi\ |       |   \14/|            |   \14/|        |   /    2/pi\      2/pi\ |
- -- - ---- - -- + -- + ---- + -- + ---- + ---- - I*log\-\/ -1 / - I*log\-(-1)   / + -pi - I*log|  /  cos |--| + sin |--| | + atan|-------| + pi - atan|-------| - I*log|  /  cos |--| + sin |--| | + -pi - I*log|  /  cos |--| + sin |--| | + atan|-------| + pi - atan|-------| - I*log|  /  cos |--| + sin |--| |
  3     7     9    9     7     3     9      7                                                   \\/       \18/       \18/ /       |   /pi\|            |   /pi\|        \\/       \18/       \18/ /              \\/       \14/       \14/ /       |   /pi\|            |   /pi\|        \\/       \14/       \14/ /
                                                                                                                                  |sin|--||            |sin|--||                                                                                   |sin|--||            |sin|--||                                   
                                                                                                                                  \   \18//            \   \18//                                                                                   \   \14//            \   \14//

$$\left(\left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)}\right) + \left(\left(\left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)}\right) + \left(- i \log{\left(- \left(-1\right)^{\frac{2}{9}} \right)} + \left(- i \log{\left(- \sqrt[7]{-1} \right)} + \left(\left(\left(\left(\left(\left(\left(- \frac{\pi}{3} - \frac{2 \pi}{7}\right) - \frac{\pi}{9}\right) + \frac{\pi}{9}\right) + \frac{2 \pi}{7}\right) + \frac{\pi}{3}\right) + \frac{7 \pi}{9}\right) + \frac{6 \pi}{7}\right)\right)\right)\right) + \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \pi\right)\right)\right) + \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \pi\right)$$

                                                   /    _____________________\          /    _____________________\
103*pi        / 7 ____\        /     2/9\          |   /    2/pi\      2/pi\ |          |   /    2/pi\      2/pi\ |
------ - I*log\-\/ -1 / - I*log\-(-1)   / - 2*I*log|  /  cos |--| + sin |--| | - 2*I*log|  /  cos |--| + sin |--| |
  63                                               \\/       \14/       \14/ /          \\/       \18/       \18/ /

$$- i \log{\left(- \sqrt[7]{-1} \right)} - i \log{\left(- \left(-1\right)^{\frac{2}{9}} \right)} - 2 i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} - 2 i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \frac{103 \pi}{63}$$

product

                                                                          /                                             /   /pi\\\ /         /   /pi\\                                   \ /                                             /   /pi\\\ /         /   /pi\\                                   \
                                                                          |           /    _____________________\       |cos|--||| |         |cos|--||        /    _____________________\| |           /    _____________________\       |cos|--||| |         |cos|--||        /    _____________________\|
-pi  -2*pi -pi  pi 2*pi pi 7*pi 6*pi /      / 7 ____\\ /      /     2/9\\ |           |   /    2/pi\      2/pi\ |       |   \18/|| |         |   \18/|        |   /    2/pi\      2/pi\ || |           |   /    2/pi\      2/pi\ |       |   \14/|| |         |   \14/|        |   /    2/pi\      2/pi\ ||
----*-----*----*--*----*--*----*----*\-I*log\-\/ -1 //*\-I*log\-(-1)   //*|-pi - I*log|  /  cos |--| + sin |--| | + atan|-------||*|pi - atan|-------| - I*log|  /  cos |--| + sin |--| ||*|-pi - I*log|  /  cos |--| + sin |--| | + atan|-------||*|pi - atan|-------| - I*log|  /  cos |--| + sin |--| ||
 3     7    9   9   7   3   9    7                                        |           \\/       \18/       \18/ /       |   /pi\|| |         |   /pi\|        \\/       \18/       \18/ /| |           \\/       \14/       \14/ /       |   /pi\|| |         |   /pi\|        \\/       \14/       \14/ /|
                                                                          |                                             |sin|--||| |         |sin|--||                                   | |                                             |sin|--||| |         |sin|--||                                   |
                                                                          \                                             \   \18/// \         \   \18//                                   / \                                             \   \14/// \         \   \14//                                   /

$$- i \log{\left(- \left(-1\right)^{\frac{2}{9}} \right)} - i \log{\left(- \sqrt[7]{-1} \right)} \frac{6 \pi}{7} \frac{7 \pi}{9} \frac{\pi}{3} \frac{2 \pi}{7} \frac{\pi}{9} \cdot - \frac{\pi}{9} \cdot - \frac{\pi}{3} \left(- \frac{2 \pi}{7}\right) \left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)}\right) \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{18} \right)}}{\sin{\left(\frac{\pi}{18} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{18} \right)} + \cos^{2}{\left(\frac{\pi}{18} \right)}} \right)} + \pi\right) \left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)}\right) \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{14} \right)}}{\sin{\left(\frac{\pi}{14} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{14} \right)} + \cos^{2}{\left(\frac{\pi}{14} \right)}} \right)} + \pi\right)$$

       12    / 7 ____\    /     2/9\
3200*pi  *log\-\/ -1 /*log\-(-1)   /
------------------------------------
             425329947

$$\frac{3200 \pi^{12} \log{\left(- \sqrt[7]{-1} \right)} \log{\left(- \left(-1\right)^{\frac{2}{9}} \right)}}{425329947}$$

3200*pi^12*log(-(-1)^(1/7))*log(-(-1)^(2/9))/425329947

Numerical answer [src]

x1 = 45.0294947014537

x2 = -93.8987137572949

x3 = -71.8078320820524

x4 = 91.5549859046168

x5 = -35.9537825910832

x6 = -91.8043186549017

x7 = 42.2369678982628

x8 = -9.87357691128221

x9 = 40.1425727958696

x10 = 49.367884556411

x11 = -27.8255349317953

x12 = -70.0126362800011

x13 = 63.8790506229925

x14 = -24.0855436775217

x15 = 35.9537825910832

x16 = 98.0875039620813

x17 = 0.349065850398866

x18 = 86.2192650485199

x19 = -8.02851455917392

x20 = -33.85938748869

x21 = -21.5423496246157

x22 = -45.7276264022514

x23 = -86.2192650485199

x24 = -95.9931088596881

x25 = -31.7649923862968

x26 = -55.6510698635906

x27 = -65.2753140245879

x28 = 44.331363000656

x29 = 70.0126362800011

x30 = -83.4766047953859

x31 = -24.2351433276927

x32 = -87.6155284501153

x33 = 17.9519580205131

x34 = -38.0481776934764

x35 = 56.1996019142174

x36 = 71.8078320820524

x37 = 12.2173047639603

x38 = 58.2939970166106

x39 = 95.9931088596881

x40 = 34.1087202389749

x41 = 29.6705972839036

x42 = -49.9164166070378

x43 = -5.93411945678072

x44 = 49.9164166070378

x45 = 16.1567622184618

x46 = -68.0678408277789

x47 = -75.7472895365539

x48 = 10.1229096615671

x49 = -26.030339129744

x50 = 111.302139727181

x51 = 38.0481776934764

x52 = -42.2369678982628

x53 = 79.9360797413403

x54 = 85.2718005974372

x55 = -53.8558740615393

x56 = -11.6687727133335

x57 = -52.010811709431

x58 = -77.1435529381494

x59 = -82.0304748437335

x60 = -168.598805742652

x61 = 27.5762021815104

x62 = 61.9342551707702

x63 = -60.1390593687189

x64 = 5.93411945678072

x65 = 68.2174404779498

x66 = -97.8381712117964

x67 = 60.1390593687189

x68 = 14.3116998663535

x69 = -13.4639685153848

x70 = -43.6332312998582

x71 = 22.4399475256414

x72 = 54.1052068118242

x73 = 19.8967534727354

x74 = 78.091017389232

x75 = -89.7597901025655

x76 = -63.7294509728215

x77 = -16.1567622184618

x78 = 74.3510261349584

x79 = -61.7846555205993

x80 = 76.2958215871807

x81 = -77.8416846389471

x82 = 32.3135244369236

x83 = -1.74532925199433

x84 = 84.1248699461267

x85 = 24.2351433276927

x86 = -19.7471538225644

x87 = -73.6528944341607

x88 = -29.6705972839036

x89 = 100.181899064475

x90 = -47.8220215046446

x91 = 93.8987137572949

x92 = 8.02851455917392

x93 = 88.3136601509131

x94 = 82.0304748437335

x95 = -99.6333670138477

x96 = 52.010811709431

x97 = -3.83972435438753

x98 = 27.8255349317953

x99 = 66.4222446758985

x99 = 66.4222446758985

Other calculators

How to use it?

Identical expressions

cos4xcos2xcosx=0,125 equation

Numerical solution:

The solution