Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+4*x+4=0
-
Equation 90+x-4*10=60
-
Equation 6*3+x=72
-
Equation x^3+x^2=0
- Express {x} in terms of y where:
- -4*x+19*y=-13
- -5*x-6*y=-1
- -16*x-2*y=4
- -7*x+15*y=18
- Derivative of:
-
cos2x
-
1/2
- Integral of d{x}:
-
cos2x
-
1/2
- Graphing y =:
- cos2x
- Sum of series:
-
1/2
-
Identical expressions
- cos two x= one /2
- co sinus of e of 2x equally 1 divide by 2
- co sinus of e of two x equally one divide by 2
- cos2x=1 divide by 2
-
Similar expressions
- sin(2*x)+cos(2*x)=-1
- cos(2*x)=-1
- 2*cos(2*x)+1=0
cos2x=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(2 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
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:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{6}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{3}$$
$$\cos{\left(2 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
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
x_1 = --
6
$$x_{1} = \frac{\pi}{6}$$
5*pi
x_2 = ----
6
$$x_{2} = \frac{5 \pi}{6}$$
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 6 6
$$\left(\frac{\pi}{6}\right) + \left(\frac{5 \pi}{6}\right)$$
=
pi
$$\pi$$
product
pi 5*pi -- * ---- 6 6
$$\left(\frac{\pi}{6}\right) * \left(\frac{5 \pi}{6}\right)$$
=
2 5*pi ----- 36
$$\frac{5 \pi^{2}}{36}$$
Numerical answer
[src]
x1 = -18.3259571459405
x2 = 60.2138591938044
x3 = -12.0427718387609
x4 = 84.2994028713261
x5 = 3.66519142918809
x6 = -2.61799387799149
x7 = -100.007366139275
x8 = -71.733032256967
x9 = 75.9218224617533
x10 = -84.2994028713261
x11 = 47.6474885794452
x12 = -82.2050077689329
x13 = -41.3643032722656
x14 = 82.2050077689329
x15 = 93.7241808320955
x16 = 97.9129710368819
x17 = -90.5825881785057
x18 = 66.497044500984
x19 = -56.025068989018
x20 = 22.5147473507269
x21 = 52.8834763354282
x22 = -65.4498469497874
x23 = -68.5914396033772
x24 = -24.60914245312
x25 = 31.9395253114962
x26 = 53.9306738866248
x27 = 0.523598775598299
x28 = 40.317105721069
x29 = 2.61799387799149
x30 = -19.3731546971371
x31 = 131.423292675173
x32 = -63.3554518473942
x33 = 85.3466004225227
x34 = 34.0339204138894
x35 = -91.6297857297023
x36 = 49.7418836818384
x37 = -81.1578102177363
x38 = -46.6002910282486
x39 = -31.9395253114962
x40 = -5.75958653158129
x41 = -16.2315620435473
x42 = -1152.44090509186
x43 = 16.2315620435473
x44 = 71.733032256967
x45 = 19.3731546971371
x46 = 25.6563400043166
x47 = 88.4881930761125
x48 = 38.2227106186758
x49 = 78.0162175641465
x50 = -47.6474885794452
x51 = 44.5058959258554
x52 = -93.7241808320955
x53 = -40.317105721069
x54 = 91.6297857297023
x55 = -25.6563400043166
x56 = 96.8657734856853
x57 = -78.0162175641465
x58 = 24.60914245312
x59 = 8.90117918517108
x60 = -35.081117965086
x61 = 12.0427718387609
x62 = 9.94837673636768
x63 = -75.9218224617533
x64 = -2685.53812004368
x65 = -7272.26339428477
x66 = -27.7507351067098
x67 = -62.3082542961976
x68 = 5.75958653158129
x69 = -53.9306738866248
x70 = -87.4409955249159
x71 = 74.8746249105567
x72 = 63.3554518473942
x73 = 68.5914396033772
x74 = -217.293491873294
x75 = -9.94837673636768
x76 = -3.66519142918809
x77 = -38.2227106186758
x78 = -21.4675497995303
x79 = 69.6386371545737
x80 = -60.2138591938044
x81 = 90.5825881785057
x82 = 30.8923277602996
x83 = 41.3643032722656
x84 = 125.140107367993
x85 = -97.9129710368819
x86 = 62.3082542961976
x87 = 18.3259571459405
x88 = 46.6002910282486
x89 = -13.0899693899575
x90 = -34.0339204138894
x91 = -49.7418836818384
x92 = -69.6386371545737
x93 = -43.4586983746588
x94 = 27.7507351067098
x95 = 100.007366139275
x96 = 56.025068989018
x97 = -85.3466004225227
x97 = -85.3466004225227
The graph