Mister Exam
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Equation sin(x)=-sqrt(2)/2
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Equation 5.6-3(2-0.4x)=0.4(4x+1)
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Identical expressions
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Similar expressions
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-sin^2x-cos^2x-2sinx=0 equation
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The solution
You have entered
[src]
2 2 - sin (x) - cos (x) - 2*sin(x) = 0
$$\left(- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\left(- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} = 0$$
transform
$$- 2 \sin{\left(x \right)} - 1 = 0$$
$$- 2 \sin{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Move free summands (without w)
from left part to right part, we given:
$$- 2 w = 1$$
Divide both parts of the equation by -2
We get the answer: w = -1/2
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$\left(- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} = 0$$
transform
$$- 2 \sin{\left(x \right)} - 1 = 0$$
$$- 2 \sin{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Move free summands (without w)
from left part to right part, we given:
$$- 2 w = 1$$
Divide both parts of the equation by -2
w = 1 / (-2)
We get the answer: w = -1/2
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
Sum and product of roots
[src]
sum
5*pi pi - ---- - -- 6 6
$$- \frac{5 \pi}{6} - \frac{\pi}{6}$$
=
-pi
$$- \pi$$
product
-5*pi -pi -----*---- 6 6
$$- \frac{5 \pi}{6} \left(- \frac{\pi}{6}\right)$$
=
2 5*pi ----- 36
$$\frac{5 \pi^{2}}{36}$$
5*pi^2/36
Rapid solution
[src]
-5*pi
x1 = -----
6
$$x_{1} = - \frac{5 \pi}{6}$$
-pi
x2 = ----
6
$$x_{2} = - \frac{\pi}{6}$$
x2 = -pi/6
Numerical answer
[src]
x1 = 68.5914396033772
x2 = -63.3554518473942
x3 = 43.4586983746588
x4 = -101.054563690472
x5 = -40.317105721069
x6 = -96.8657734856853
x7 = -31.9395253114962
x8 = 35.081117965086
x9 = 437.20497762458
x10 = -2.61799387799149
x11 = 41.3643032722656
x12 = 12.0427718387609
x13 = 81.1578102177363
x14 = 87.4409955249159
x15 = -65.4498469497874
x16 = -57.0722665402146
x17 = 37.1755130674792
x18 = -19.3731546971371
x19 = -69.6386371545737
x20 = 97.9129710368819
x21 = -34.0339204138894
x22 = 16.2315620435473
x23 = -21.4675497995303
x24 = -78.0162175641465
x25 = -75.9218224617533
x26 = 85.3466004225227
x27 = -90.5825881785057
x28 = 5.75958653158129
x29 = 72.7802298081635
x30 = 9.94837673636768
x31 = 60.2138591938044
x32 = 56.025068989018
x33 = -284109.408028617
x34 = -94.7713783832921
x35 = 79.0634151153431
x36 = -88.4881930761125
x37 = 91.6297857297023
x38 = 100.007366139275
x39 = 192.160750644576
x40 = -13.0899693899575
x41 = 49.7418836818384
x42 = 80373.9828506657
x43 = -38.2227106186758
x44 = -82.2050077689329
x45 = 24.60914245312
x46 = 66.497044500984
x47 = -59.1666616426078
x48 = -15.1843644923507
x49 = -27.7507351067098
x50 = -52.8834763354282
x51 = -71.733032256967
x52 = -44.5058959258554
x53 = -46.6002910282486
x54 = 22.5147473507269
x55 = -151.320046147908
x56 = 53.9306738866248
x57 = -195.302343298165
x58 = -0.523598775598299
x59 = 47.6474885794452
x60 = -84.2994028713261
x61 = 18.3259571459405
x62 = -25.6563400043166
x63 = -50.789081233035
x64 = 3.66519142918809
x65 = 30.8923277602996
x66 = -6.80678408277789
x67 = -8.90117918517108
x68 = 28.7979326579064
x69 = 93.7241808320955
x70 = 62.3082542961976
x71 = 74.8746249105567
x71 = 74.8746249105567