Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^3-x^2-x-1=0
-
Equation x-6/2-x/3=3
- Equation (x+20)*(-x+10)=0
-
Equation cos(x)-sin(x)=1
- Express {x} in terms of y where:
- 15*x+7*y=2
- 12*x-20*y=10
- 6*x-2*y=20
- -11*x+10*y=-3
-
Identical expressions
- cos(x+(two *pi)/ three)= zero
- co sinus of e of (x plus (2 multiply by Pi ) divide by 3) equally 0
- co sinus of e of (x plus (two multiply by Pi ) divide by three) equally zero
- cos(x+(2pi)/3)=0
- cosx+2pi/3=0
- cos(x+(2*pi)/3)=O
- cos(x+(2*pi) divide by 3)=0
-
Similar expressions
- cos(x-(2*pi)/3)=0
cos(x+(2*pi)/3)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ 2*pi\ cos|x + ----| = 0 \ 3 /
$$\cos{\left(x + \frac{2 \pi}{3} \right)} = 0$$
Detail solution
Given the equation
$$\cos{\left(x + \frac{2 \pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(x + \frac{2 \pi}{3} \right)} = 0$$
The equation is transformed to
$$\sin{\left(x + \frac{\pi}{6} \right)} = 0$$
This equation is transformed to
$$x + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x + \frac{\pi}{6} = 2 \pi n$$
$$x + \frac{\pi}{6} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
$$\cos{\left(x + \frac{2 \pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\cos{\left(x + \frac{2 \pi}{3} \right)} = 0$$
Divide both parts of the equation by -1
The equation is transformed to
$$\sin{\left(x + \frac{\pi}{6} \right)} = 0$$
This equation is transformed to
$$x + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x + \frac{\pi}{6} = 2 \pi n$$
$$x + \frac{\pi}{6} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
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
$$- \frac{\pi}{6} + \frac{5 \pi}{6}$$
=
2*pi ---- 3
$$\frac{2 \pi}{3}$$
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 = 71.733032256967
x2 = 78.0162175641465
x3 = -57.0722665402146
x4 = -79.0634151153431
x5 = 81.1578102177363
x6 = 100.007366139275
x7 = -25.6563400043166
x8 = -31.9395253114962
x9 = -60.2138591938044
x10 = -53.9306738866248
x11 = -97.9129710368819
x12 = -820.479281362534
x13 = -72.7802298081635
x14 = 90.5825881785057
x15 = 131.423292675173
x16 = -75.9218224617533
x17 = -82.2050077689329
x18 = 93.7241808320955
x19 = -19.3731546971371
x20 = 84.2994028713261
x21 = 59.1666616426078
x22 = -69.6386371545737
x23 = -16.2315620435473
x24 = -101.054563690472
x25 = -22.5147473507269
x26 = -13.0899693899575
x27 = 40.317105721069
x28 = -44.5058959258554
x29 = -28.7979326579064
x30 = -41.3643032722656
x31 = 68.5914396033772
x32 = 24.60914245312
x33 = 18.3259571459405
x34 = -6.80678408277789
x35 = -85.3466004225227
x36 = -88.4881930761125
x37 = 56.025068989018
x38 = -47.6474885794452
x39 = -9.94837673636768
x40 = -91.6297857297023
x41 = 46.6002910282486
x42 = -3.66519142918809
x43 = 37.1755130674792
x44 = 96.8657734856853
x45 = 34.0339204138894
x46 = 12.0427718387609
x47 = 5.75958653158129
x48 = -66.497044500984
x49 = -50.789081233035
x50 = -0.523598775598299
x51 = 43.4586983746588
x52 = 30.8923277602996
x53 = 49.7418836818384
x54 = -35.081117965086
x55 = -63.3554518473942
x56 = 52.8834763354282
x57 = 74.8746249105567
x58 = -94.7713783832921
x59 = 2.61799387799149
x60 = 65.4498469497874
x61 = 62.3082542961976
x62 = 8.90117918517108
x63 = 87.4409955249159
x64 = 21.4675497995303
x65 = -38.2227106186758
x66 = 27.7507351067098
x67 = 15.1843644923507
x67 = 15.1843644923507