Mister Exam
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Similar expressions
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sinx*(x+1)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
transform
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand brackets in the left part
Divide both parts of the equation by 1 + x
We get the answer: w = 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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{2} = 2 \pi n + \pi$$
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
transform
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
$$\left(x + 1\right) \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Expand brackets in the left part
wx+1 = 0
Divide both parts of the equation by 1 + x
w = 0 / (1 + x)
We get the answer: w = 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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{2} = 2 \pi n + \pi$$
Sum and product of roots
[src]
sum
-1 + pi
$$-1 + \pi$$
=
-1 + pi
$$-1 + \pi$$
product
-0*pi
$$- 0 \pi$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = -43.9822971502571
x2 = -31.4159265358979
x3 = 84.8230016469244
x4 = -91.106186954104
x5 = -1.0
x6 = 91.106186954104
x7 = -97.3893722612836
x8 = 6.28318530717959
x9 = -72.2566310325652
x10 = -47.1238898038469
x11 = 94.2477796076938
x12 = 50.2654824574367
x13 = 56.5486677646163
x14 = 43.9822971502571
x15 = 47.1238898038469
x16 = -50.2654824574367
x17 = 37.6991118430775
x18 = -28.2743338823081
x19 = 65.9734457253857
x20 = 15.707963267949
x21 = 28.2743338823081
x22 = -62.8318530717959
x23 = 40.8407044966673
x24 = -40.8407044966673
x25 = -6.28318530717959
x26 = -81.6814089933346
x27 = -15.707963267949
x28 = -59.6902604182061
x29 = 72.2566310325652
x30 = 3.14159265358979
x31 = -25.1327412287183
x32 = 21.9911485751286
x33 = -75.398223686155
x34 = -56.5486677646163
x35 = -69.1150383789755
x36 = -84.8230016469244
x37 = 78.5398163397448
x38 = 9.42477796076938
x39 = -103.672557568463
x40 = -53.4070751110265
x41 = 62.8318530717959
x42 = -18.8495559215388
x43 = 25.1327412287183
x44 = 100.530964914873
x45 = -87.9645943005142
x46 = -9.42477796076938
x47 = 75.398223686155
x48 = 81.6814089933346
x49 = 87.9645943005142
x50 = 12.5663706143592
x51 = -34.5575191894877
x52 = 69.1150383789755
x53 = -3.14159265358979
x54 = 0.0
x55 = -21.9911485751286
x56 = -37.6991118430775
x57 = 31.4159265358979
x58 = -78.5398163397448
x59 = -12.5663706143592
x60 = -94.2477796076938
x61 = 97.3893722612836
x62 = -100.530964914873
x63 = 59.6902604182061
x64 = 53.4070751110265
x65 = 34.5575191894877
x66 = -65.9734457253857
x67 = 18.8495559215388
x67 = 18.8495559215388