Mister Exam
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How to use it?
- Equation:
-
Equation cos(8x)=-(1/2)
- Equation (13+x)×12=80
- Equation (3y+2)/(4y^2+y)+(y-3)/(16y^2-1)=3/(4y-1)
-
Equation x^4−13x^2+36=0.
- Express {x} in terms of y where:
- 5*x-5*y=20
- 8*x-7*y=-13
- 15*x-15*y=16
- 9*x+18*y=6
- Integral of d{x}:
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cos(8x)
-
Identical expressions
- cos(8x)=-(one / two)
- co sinus of e of (8x) equally minus (1 divide by 2)
- co sinus of e of (8x) equally minus (one divide by two)
- cos8x=-1/2
- cos(8x)=-(1 divide by 2)
-
Similar expressions
- cos(8*x)+7*sin(4*x)-6=0
- sin6x+√3cos6x=-2cos8x.
- sinx=-1/2
- log(3)(-cosx)+log1/3(-sinx)=-1/2
- cos(8x)=+(1/2)
- cos(8*x/5)=0
- sin(6*x)+cos(6*x)*sqrt(3)=-2*cos(8*x)
- log2(56)-(1/2log2(49))=x
- sqrt((2-x)/(x-1))-7(sqrt((x-1)/(2-x)))=6
cos(8x)=-(1/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(8 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$8 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$8 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$8 x = \pi n + \frac{2 \pi}{3}$$
$$8 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$8$$
we get the answer:
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{24}$$
$$\cos{\left(8 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$8 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$8 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$8 x = \pi n + \frac{2 \pi}{3}$$
$$8 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$8$$
we get the answer:
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{24}$$
Sum and product of roots
[src]
sum
pi pi -- + -- 12 6
$$\left(\frac{\pi}{12}\right) + \left(\frac{\pi}{6}\right)$$
=
pi -- 4
$$\frac{\pi}{4}$$
product
pi pi -- * -- 12 6
$$\left(\frac{\pi}{12}\right) * \left(\frac{\pi}{6}\right)$$
=
2 pi --- 72
$$\frac{\pi^{2}}{72}$$
Rapid solution
[src]
pi
x_1 = --
12
$$x_{1} = \frac{\pi}{12}$$
pi
x_2 = --
6
$$x_{2} = \frac{\pi}{6}$$
Numerical answer
[src]
x1 = 24.0855436775217
x2 = 12.30457122656
x3 = 44.2440965380563
x4 = 60.2138591938044
x5 = 96.0803753222878
x6 = -12.0427718387609
x7 = 84.2994028713261
x8 = -1.83259571459405
x9 = -90.8443875663049
x10 = -100.007366139275
x11 = -71.733032256967
x12 = 28.012534494509
x13 = 75.9218224617533
x14 = -17.8023583703422
x15 = 90.0589894029074
x16 = 92.1533845053006
x17 = 82.2050077689329
x18 = 97.9129710368819
x19 = 66.2352451131848
x20 = 71.9948316447661
x21 = -56.025068989018
x22 = -15.9697626557481
x23 = -37.9609112308767
x24 = -41.8879020478639
x25 = 8.11578102177363
x26 = -63.8790506229925
x27 = 31.9395253114962
x28 = 53.9306738866248
x29 = -39.7935069454707
x30 = 40.317105721069
x31 = -45.8148928648512
x32 = -35.8665161284835
x33 = -87.7027949127151
x34 = 20.1585528605345
x35 = -78.801615727544
x36 = -5.75958653158129
x37 = -67.8060414399797
x38 = 34.2957198016886
x39 = -85.870199198121
x40 = -11.2573736753634
x41 = 64.1408500107916
x42 = 16.2315620435473
x43 = 68.0678408277789
x44 = 26.1799387799149
x45 = 38.2227106186758
x46 = 56.2868683768171
x47 = -79.8488132787406
x48 = -83.7758040957278
x49 = 139.539073696947
x50 = -93.7241808320955
x51 = -34.2957198016886
x52 = -52.3598775598299
x53 = -78.0162175641465
x54 = -57.857664703612
x55 = 88.2263936883134
x56 = 2.0943951023932
x57 = -19.8967534727354
x58 = 9.94837673636768
x59 = 86.1319985859202
x60 = -27.7507351067098
x61 = -96.0803753222878
x62 = -61.7846555205993
x63 = 46.0766922526503
x64 = 52.0980781720307
x65 = -25.3945406165175
x66 = -65.7116463375865
x67 = 74.0892267471593
x68 = -81.9432083811338
x69 = -43.720497762458
x70 = 30.1069295969022
x71 = -3.66519142918809
x72 = 78.2780169519457
x73 = 6.02138591938044
x74 = 70.162235930172
x75 = -59.9520598060052
x76 = -89.7971900151083
x77 = -21.7293491873294
x78 = 0.261799387799149
x79 = -97.9129710368819
x80 = 93.9859802198946
x81 = 62.3082542961976
x82 = 18.3259571459405
x83 = 50.0036830696375
x84 = -13.8753675533549
x85 = 100.269165527074
x86 = -23.8237442897226
x87 = -34.0339204138894
x88 = 48.1710873550435
x89 = -49.7418836818384
x90 = 22.2529479629277
x91 = 4.18879020478639
x92 = 42.1497014356631
x93 = 81.9432083811338
x93 = 81.9432083811338
The graph