Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 8*x-3*(2*x+1)=2*x+4
-
Equation 3*9^(x-1/2)-7*6^x+3*4^(x+1)=0
-
Equation 9x-26=30-5x
-
Equation x+19=12
- Express {x} in terms of y where:
- 12*x+14*y=-15
- -9*x-15*y=-4
- 12*x-14*y=20
- 11*x-20*y=-10
-
Identical expressions
- sin(x)- one / two = zero
- sinus of (x) minus 1 divide by 2 equally 0
- sinus of (x) minus one divide by two equally zero
- sinx-1/2=0
- sin(x)-1/2=O
- sin(x)-1 divide by 2=0
-
Similar expressions
- cos(x)*sin(x)+sqrt(2)/2*cos(x)-sqrt(2)/2*sin(x)-(1/2)=0
- sin(x)+1/2=0
- x=-2*sin(x)-(1/2)
- x=1+(sin(x-(1/2)))/2
- cos(x)sqr(sinx-1/2)=0
- sinx-1/2=0
sin(x)-1/2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(x \right)} - \frac{1}{2} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x \right)} = \frac{1}{2}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n + \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
$$\sin{\left(x \right)} - \frac{1}{2} = 0$$
- this is the simplest trigonometric equation
Move -1/2 to right part of the equation
with the change of sign in -1/2
We get:
$$\sin{\left(x \right)} = \frac{1}{2}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n + \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
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 = -79.0634151153431
x2 = -49.7418836818384
x3 = -87.4409955249159
x4 = -41.3643032722656
x5 = -74.8746249105567
x6 = -9.94837673636768
x7 = -28.7979326579064
x8 = -68.5914396033772
x9 = 15.1843644923507
x10 = 17438.4572213013
x11 = -100.007366139275
x12 = -97.9129710368819
x13 = -35.081117965086
x14 = 2.61799387799149
x15 = 88.4881930761125
x16 = -56.025068989018
x17 = -47.6474885794452
x18 = 94.7713783832921
x19 = 63.3554518473942
x20 = -43.4586983746588
x21 = 69.6386371545737
x22 = 50.789081233035
x23 = -62.3082542961976
x24 = -30.8923277602996
x25 = 57.0722665402146
x26 = -53.9306738866248
x27 = 59.1666616426078
x28 = 44.5058959258554
x29 = -12.0427718387609
x30 = 101.054563690472
x31 = 82.2050077689329
x32 = 38.2227106186758
x33 = -81.1578102177363
x34 = 46.6002910282486
x35 = 27.7507351067098
x36 = -18.3259571459405
x37 = 138.753675533549
x38 = 96.8657734856853
x39 = -93.7241808320955
x40 = 75.9218224617533
x41 = -16.2315620435473
x42 = 78.0162175641465
x43 = -37.1755130674792
x44 = 84.2994028713261
x45 = 6.80678408277789
x46 = 31.9395253114962
x47 = -4454.25478401473
x48 = 34.0339204138894
x49 = 13.0899693899575
x50 = 0.523598775598299
x51 = -22.5147473507269
x52 = -66.497044500984
x53 = 40.317105721069
x54 = -2650.98060085419
x55 = -627.79493194236
x56 = -3.66519142918809
x57 = 21.4675497995303
x58 = -91.6297857297023
x59 = 8.90117918517108
x60 = 52.8834763354282
x61 = 25.6563400043166
x62 = -72.7802298081635
x63 = -60.2138591938044
x64 = -24.60914245312
x65 = 65.4498469497874
x66 = 134.564885328763
x67 = 90.5825881785057
x68 = -5.75958653158129
x69 = -85.3466004225227
x70 = 71.733032256967
x71 = 19.3731546971371
x71 = 19.3731546971371
The graph