Mister Exam
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How to use it?
- Equation:
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Equation sin(x)=1/5
-
Equation x^2+11=0
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Equation x^2+5x+6=0
-
Equation -x/7+x=15/7
- Express {x} in terms of y where:
- 9*x+17*y=13
- 20*x+18*y=-9
- -4*x-8*y=-20
- -1*x+4*y=-10
-
Identical expressions
- sinx(2sinx- one)= zero
- sinus of x(2 sinus of x minus 1) equally 0
- sinus of x(2 sinus of x minus one) equally zero
- sinx2sinx-1=0
- sinx(2sinx-1)=O
-
Similar expressions
- sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)+sin(4*pi/3)=0
- sinx(2sinx+1)=0
sinx(2sinx-1)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
sin(x)*(2*sin(x) - 1) = 0
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
transform
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand the expression in the equation
$$w \left(2 w - 1\right) = 0$$
We get the quadratic equation
$$2 w^{2} - w = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 2$$
$$b = -1$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
transform
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
$$\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand the expression in the equation
$$w \left(2 w - 1\right) = 0$$
We get the quadratic equation
$$2 w^{2} - w = 0$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 2$$
$$b = -1$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (2) * (0) = 1
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
pi
x2 = --
6
$$x_{2} = \frac{\pi}{6}$$
5*pi
x3 = ----
6
$$x_{3} = \frac{5 \pi}{6}$$
x4 = pi
$$x_{4} = \pi$$
x4 = pi
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- + pi 6 6
$$\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \pi$$
=
2*pi
$$2 \pi$$
product
pi 5*pi 0*--*----*pi 6 6
$$\pi \frac{5 \pi}{6} \cdot 0 \frac{\pi}{6}$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 25.1327412287183
x2 = 18.8495559215388
x3 = 94.2477796076938
x4 = -56.025068989018
x5 = -47.6474885794452
x6 = 65.9734457253857
x7 = -34.5575191894877
x8 = -84.8230016469244
x9 = 38.2227106186758
x10 = -15.707963267949
x11 = 75.9218224617533
x12 = 78.0162175641465
x13 = 6.28318530717959
x14 = 84.2994028713261
x15 = -31.4159265358979
x16 = -87.9645943005142
x17 = -72.2566310325652
x18 = -37.6991118430775
x19 = -143.989663289532
x20 = 128.281700021583
x21 = 12.5663706143592
x22 = -49.7418836818384
x23 = 72.2566310325652
x24 = -41.3643032722656
x25 = -68.5914396033772
x26 = -100.007366139275
x27 = 69.6386371545737
x28 = -106.290551446455
x29 = 34.0339204138894
x30 = 62.8318530717959
x31 = 53.4070751110265
x32 = -25.1327412287183
x33 = 100.530964914873
x34 = 0.0
x35 = -65.9734457253857
x36 = -21.9911485751286
x37 = 90.5825881785057
x38 = 97.3893722612836
x39 = -78.5398163397448
x40 = -74.8746249105567
x41 = -97.9129710368819
x42 = 2.61799387799149
x43 = 50.2654824574367
x44 = 43.9822971502571
x45 = -62.3082542961976
x46 = -53.4070751110265
x47 = -18.8495559215388
x48 = -62.8318530717959
x49 = -93.7241808320955
x50 = 15.707963267949
x51 = 28.2743338823081
x52 = 21.9911485751286
x53 = 40.317105721069
x54 = 3.14159265358979
x55 = -91.6297857297023
x56 = 91.106186954104
x57 = -85.3466004225227
x58 = 59.6902604182061
x59 = -28.2743338823081
x60 = -5.75958653158129
x61 = 46.6002910282486
x62 = -476.99848457005
x63 = -59.6902604182061
x64 = 69.1150383789755
x65 = -40.8407044966673
x66 = -9.94837673636768
x67 = 56.5486677646163
x68 = -69.1150383789755
x69 = -81.6814089933346
x70 = 50.789081233035
x71 = 9.42477796076938
x72 = -53.9306738866248
x73 = 87.9645943005142
x74 = -43.9822971502571
x75 = -12.0427718387609
x76 = 82.2050077689329
x77 = 96.8657734856853
x78 = 31.9395253114962
x79 = -75.398223686155
x80 = 47.1238898038469
x81 = -3.66519142918809
x82 = 25.6563400043166
x83 = -24.60914245312
x84 = -18.3259571459405
x85 = 19.3731546971371
x85 = 19.3731546971371
The graph