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3–sin^2x–3cosx=0 equation
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The solution
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[src]
2 3 - sin (x) - 3*cos(x) = 0
$$\left(3 - \sin^{2}{\left(x \right)}\right) - 3 \cos{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\left(3 - \sin^{2}{\left(x \right)}\right) - 3 \cos{\left(x \right)} = 0$$
transform
$$\cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 2 = 0$$
$$\cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 2 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = 1$$
$$b = -3$$
$$c = 2$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 2$$
$$w_{2} = 1$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
$$\left(3 - \sin^{2}{\left(x \right)}\right) - 3 \cos{\left(x \right)} = 0$$
transform
$$\cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 2 = 0$$
$$\cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 2 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = 1$$
$$b = -3$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (2) = 1
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 2$$
$$w_{2} = 1$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(2 \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
Sum and product of roots
[src]
sum
/ ___\ / ___\
|\/ 3 | |\/ 3 |
- 2*I*atanh|-----| + 2*I*atanh|-----|
\ 3 / \ 3 /
$$- 2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)} + 2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)}$$
=
0
$$0$$
product
/ ___\ / ___\
|\/ 3 | |\/ 3 |
0*-2*I*atanh|-----|*2*I*atanh|-----|
\ 3 / \ 3 /
$$2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)} 0 \left(- 2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
/ ___\
|\/ 3 |
x2 = -2*I*atanh|-----|
\ 3 /
$$x_{2} = - 2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)}$$
/ ___\
|\/ 3 |
x3 = 2*I*atanh|-----|
\ 3 /
$$x_{3} = 2 i \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)}$$
x3 = 2*i*atanh(sqrt(3)/3)
Numerical answer
[src]
x1 = -69.1150378835413
x2 = 87.9645943361855
x3 = -5215.04380391355
x4 = 25.1327418552092
x5 = -87.9645943583137
x6 = -37.6991105864418
x7 = -50.2654835458582
x8 = 43.9822984432902
x9 = 18.8495567908524
x10 = -43.9822971744631
x11 = -25.1327403745683
x12 = 6.28318657126167
x13 = -56.5486673890729
x14 = -31.4159274784001
x15 = -100.530964552074
x16 = -31.4159267343047
x17 = -87.9645929959187
x18 = 0.0
x19 = -56.548667601399
x20 = 100.530963867505
x21 = -43.9822984008141
x22 = 37.6991120523088
x23 = 18.8495555496789
x24 = -12.566370225933
x25 = -18.8495553098906
x26 = 6.28318403478594
x27 = 50.2654811803455
x28 = 31.4159264886041
x29 = -75.3982238951945
x30 = -100.530964660003
x31 = 37.6991128567789
x32 = 56.5486688750198
x33 = 75.3982241007593
x34 = -1.24358883358213e-6
x35 = 94.2477808851498
x36 = 12.5663696338539
x37 = -94.2477807087722
x38 = 50.2654837284661
x39 = -12.5663714609397
x40 = 50.2654824463236
x41 = 69.1150390203837
x42 = 81.6814099712889
x43 = 62.8318532824014
x44 = -25.1327416132853
x45 = 81.6814092136757
x46 = -62.8318524749096
x47 = 6.28318528412712
x48 = -75.3982226148333
x49 = -43.9822958485246
x50 = 18.8495560791567
x51 = -6.28318638269977
x52 = 87.9645955876053
x53 = -18.8495565271099
x54 = 62.8318527124984
x55 = -81.6814102528289
x56 = 62.8318539562208
x57 = -62.8318536920999
x58 = -69.1150387763928
x59 = 75.3982235113583
x60 = -94.2477785033657
x61 = 31.4159257044419
x62 = -75.3982245873506
x63 = 43.9822971695364
x64 = 75.398222869856
x65 = -87.9645955574957
x66 = 12.5663704264737
x67 = -56.5486686262791
x68 = -12.5663705508209
x69 = -31.4159254513224
x70 = -81.6814090386471
x71 = 56.5486667492183
x72 = -6.2831842606194
x73 = 1.29789534933499e-6
x74 = 94.2477783269892
x75 = 87.9645930723958
x76 = 31.4159261902552
x77 = 25.1327406377965
x78 = 94.2477796093521
x79 = -94.2477794313305
x80 = -6.28318510897268
x81 = 43.9822959141556
x82 = 31.4159269373448
x83 = 69.1150378026136
x84 = -25.1327409215642
x85 = 100.530964747725
x86 = -37.6991118774752
x87 = 12.5663717119664
x88 = -50.2654813807249
x89 = -81.6814077446239
x90 = 37.6991107806887
x91 = -69.1150375400315
x92 = 62.8318531044175
x93 = -50.2654822701809
x94 = -37.6991131096277
x95 = 81.6814079442934
x96 = 56.5486675871477
x96 = 56.5486675871477