Mister Exam
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2*cos(2*x)-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \cos{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(2 x \right)} = 1$$
The equation is transformed to
$$\cos{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \frac{\pi}{3}$$
$$2 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{6}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{3}$$
$$2 \cos{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move -1 to right part of the equation
with the change of sign in -1
We get:
$$2 \cos{\left(2 x \right)} = 1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \frac{\pi}{3}$$
$$2 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{6}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{3}$$
Rapid solution
[src]
pi
x1 = --
6
$$x_{1} = \frac{\pi}{6}$$
5*pi
x2 = ----
6
$$x_{2} = \frac{5 \pi}{6}$$
x2 = 5*pi/6
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 6 6
$$\frac{\pi}{6} + \frac{5 \pi}{6}$$
=
pi
$$\pi$$
product
pi 5*pi --*---- 6 6
$$\frac{\pi}{6} \frac{5 \pi}{6}$$
=
2 5*pi ----- 36
$$\frac{5 \pi^{2}}{36}$$
5*pi^2/36
Numerical answer
[src]
x1 = -35.081117965086
x2 = 88.4881930761125
x3 = -56.025068989018
x4 = -47.6474885794452
x5 = -34.0339204138894
x6 = 62.3082542961976
x7 = -40.317105721069
x8 = 38.2227106186758
x9 = 75.9218224617533
x10 = 78.0162175641465
x11 = 84.2994028713261
x12 = 49.7418836818384
x13 = 18.3259571459405
x14 = -69.6386371545737
x15 = -75.9218224617533
x16 = 100.007366139275
x17 = 44.5058959258554
x18 = 71.733032256967
x19 = -21.4675497995303
x20 = -49.7418836818384
x21 = -46.6002910282486
x22 = -87.4409955249159
x23 = 93.7241808320955
x24 = -41.3643032722656
x25 = 125.140107367993
x26 = -2.61799387799149
x27 = -68.5914396033772
x28 = -38.2227106186758
x29 = -100.007366139275
x30 = 69.6386371545737
x31 = -25.6563400043166
x32 = -16.2315620435473
x33 = 34.0339204138894
x34 = 9.94837673636768
x35 = 24.60914245312
x36 = 41.3643032722656
x37 = 52.8834763354282
x38 = 16.2315620435473
x39 = 12.0427718387609
x40 = -82.2050077689329
x41 = -27.7507351067098
x42 = 5.75958653158129
x43 = -60.2138591938044
x44 = 90.5825881785057
x45 = 66.497044500984
x46 = -84.2994028713261
x47 = -217.293491873294
x48 = 22.5147473507269
x49 = -97.9129710368819
x50 = 2.61799387799149
x51 = -43.4586983746588
x52 = -1152.44090509186
x53 = -62.3082542961976
x54 = 56.025068989018
x55 = 3.66519142918809
x56 = -7272.26339428477
x57 = -13.0899693899575
x58 = -78.0162175641465
x59 = -93.7241808320955
x60 = 68.5914396033772
x61 = 40.317105721069
x62 = 74.8746249105567
x63 = -91.6297857297023
x64 = -85.3466004225227
x65 = 91.6297857297023
x66 = 27.7507351067098
x67 = -5.75958653158129
x68 = 46.6002910282486
x69 = 85.3466004225227
x70 = -65.4498469497874
x71 = -9.94837673636768
x72 = 30.8923277602996
x73 = 131.423292675173
x74 = -19.3731546971371
x75 = 63.3554518473942
x76 = -71.733032256967
x77 = -53.9306738866248
x78 = -90.5825881785057
x79 = 47.6474885794452
x80 = -12.0427718387609
x81 = 82.2050077689329
x82 = -81.1578102177363
x83 = 96.8657734856853
x84 = 97.9129710368819
x85 = 31.9395253114962
x86 = -31.9395253114962
x87 = -2685.53812004368
x88 = 0.523598775598299
x89 = 53.9306738866248
x90 = -63.3554518473942
x91 = -3.66519142918809
x92 = 8.90117918517108
x93 = 25.6563400043166
x94 = 60.2138591938044
x95 = -24.60914245312
x96 = -18.3259571459405
x97 = 19.3731546971371
x97 = 19.3731546971371
The graph