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Sin^2x+5sinx+4=0

Sin^2x+5sinx+4=0 equation

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Numerical solution:

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The solution

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   2                      
sin (x) + 5*sin(x) + 4 = 0
sin⁡2(x)+5sin⁡(x)+4=0\sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 4 = 0
Simplify
Detail solution
Given the equation:
sin⁡2(x)+5sin⁡(x)+4=0\sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 4 = 0
Transform
sin⁡2(x)+5sin⁡(x)+4=0\sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 4 = 0
(sin⁡(x)+1)(sin⁡(x)+4)=0\left(\sin{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + 4\right) = 0
Consider each factor separately

Step


sin⁡(x)+1=0\sin{\left(x \right)} + 1 = 0
- this is the simplest trigonometric equation
Move 11 to right part of the equation
with the change of sign in 11
We get:
sin⁡(x)=−1\sin{\left(x \right)} = -1
This equation is transformed to
x=2πn+asin⁡(−1)x = 2 \pi n + \operatorname{asin}{\left(-1 \right)}
x=2πn−asin⁡(−1)+πx = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi
Or
x=2πn−π2x = 2 \pi n - \frac{\pi}{2}
x=2πn+3π2x = 2 \pi n + \frac{3 \pi}{2}
, where n - is a integer

Step


sin⁡(x)+4=0\sin{\left(x \right)} + 4 = 0
- this is the simplest trigonometric equation
Move 44 to right part of the equation
with the change of sign in 44
We get:
sin⁡(x)=−4\sin{\left(x \right)} = -4
As right part of the equation
modulo =
4>14 > 1
but sin can no be more than 1 or less than -1
so the solution of the equation d'not exist.
The final answer:
x1=2πn−π2x_{1} = 2 \pi n - \frac{\pi}{2}
x2=2πn+3π2x_{2} = 2 \pi n + \frac{3 \pi}{2}
The graph
0-80-60-40-2020406080-100100020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
      -pi 
x_1 = ----
       2
x1=−π2x_{1} = - \frac{\pi}{2}
Simplify 
      3*pi
x_2 = ----
       2
x2=3π2x_{2} = \frac{3 \pi}{2}
Simplify 
x_3 = pi + I*im(asin(4)) + re(asin(4))
x3=re⁡(asin⁡(4))+π+iim⁡(asin⁡(4))x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}
Simplify 
x_4 = -re(asin(4)) - I*im(asin(4))
x4=−re⁡(asin⁡(4))−iim⁡(asin⁡(4))x_{4} = - \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}
Simplify
Sum and product of roots [src] 
sum
-pi    3*pi                                                                  
---- + ---- + pi + I*im(asin(4)) + re(asin(4)) + -re(asin(4)) - I*im(asin(4))
 2      2
(−π2)+(3π2)+(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4)))+(−re⁡(asin⁡(4))−iim⁡(asin⁡(4)))\left(- \frac{\pi}{2}\right) + \left(\frac{3 \pi}{2}\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) 
=
2*pi
2π2 \pi
Simplify 
product
-pi    3*pi                                                                  
---- * ---- * pi + I*im(asin(4)) + re(asin(4)) * -re(asin(4)) - I*im(asin(4))
 2      2
(−π2)∗(3π2)∗(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4)))∗(−re⁡(asin⁡(4))−iim⁡(asin⁡(4)))\left(- \frac{\pi}{2}\right) * \left(\frac{3 \pi}{2}\right) * \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) * \left(- \operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) 
=
    2                                                                 
3*pi *(I*im(asin(4)) + re(asin(4)))*(pi + I*im(asin(4)) + re(asin(4)))
----------------------------------------------------------------------
                                  4
3π2(re⁡(asin⁡(4))+iim⁡(asin⁡(4)))(re⁡(asin⁡(4))+π+iim⁡(asin⁡(4)))4\frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(4 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(4 \right)}\right)}\right)}{4}
Simplify
Numerical answer [src] 
x1 = -1.57079553723028
x2 = -58.1194648756724
x3 = 61.261057169253
x4 = 29.8451303249477
x5 = -83.2522055900296
x6 = -14.1371668356852
x7 = 86.393797270252
x8 = -51.8362795864847
x9 = 92.6769830471807
x10 = -95.8185751137388
x11 = 80.1106131328523
x12 = -58.1194633663376
x13 = 42.4115001349287
x14 = -102.101760504285
x15 = 36.1283147100469
x16 = 42.4115016172874
x17 = 36.1283163337474
x18 = 48.69468600764
x19 = -76.9690194698719
x20 = -20.4203529321792
x21 = -26.7035371497841
x22 = -64.4026491302014
x23 = 48.6946858878928
x24 = -39.2699081380954
x25 = -7.85398149543083
x26 = 80.1106134819013
x27 = 92.6769839880501
x28 = -89.5353912618095
x29 = 98.9601691665388
x30 = 80.1106118591254
x31 = -45.5530941312263
x32 = 86.3937987715402
x33 = -51.8362779640963
x34 = 4.71238893038135
x35 = 23.561945050292
x36 = 54.9778720075173
x37 = 111.526538895344
x38 = -39.2699074847409
x39 = -51.8362786890529
x40 = -76.9690204292399
x41 = -1.57079643327875
x42 = 4.71238967299068
x43 = -32.986722310692
x44 = -95.8185767338788
x45 = 29.845129433396
x46 = 67.5442421304766
x47 = 98.9601682085687
x48 = -14.1371677212728
x49 = -45.5530935928574
x50 = 92.6769830923451
x51 = -95.8185758679983
x52 = 54.9778710491187
x53 = 48.6946868305897
x54 = -64.4026500897686
x55 = 73.8274274853344
x56 = -32.9867224840726
x57 = -7.85398081492663
x58 = -20.4203520369174
x59 = 23.5619442255439
x60 = 73.8274265883229
x61 = 67.5442413833945
x62 = 73.827428039942
x63 = -64.4026491499229
x64 = -26.7035381078102
x65 = -89.5353907523617
x66 = -89.5353898470206
x67 = 36.1283157342909
x68 = 4.71238872859587
x69 = -83.2522051960298
x70 = -32.9867232697875
x71 = -76.969019368271
x72 = -7.85398243849929
x73 = -1.57079699893666
x74 = 86.3937978861751
x75 = -70.6858352668671
x76 = 29.8451309058722
x77 = -70.6858343093075
x78 = -45.5530926920028
x79 = 10.995574848439
x80 = -83.2522046425991
x81 = -14.1371662295881
x82 = -58.1194639966323
x83 = -39.2699084307113
x84 = 10.9955738896661
x85 = -20.4203519904496
x86 = 1022.58840810359
x87 = 23.5619451690214
x88 = 42.4115007263916
x89 = 17.2787600097367
x90 = 67.5442423284925
x91 = 17.2787590512974
x92 = 61.2610562104516
x92 = 61.2610562104516
The graph
Sin^2x+5sinx+4=0 equation
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