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Similar expressions
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tg^2(x)+ctg^2(x)+3*(tg(x)+ctg(x))+4=0 equation
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The solution
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[src]
2 2 tan (x) + cot (x) + 3*(tan(x) + cot(x)) + 4 = 0
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 4 = 0$$
Detail solution
Given the equation
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 4 = 0$$
transform
$$\tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 \cot{\left(x \right)} + 3 = 0$$
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
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
$$a = 1$$
$$b = 3$$
$$c = \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + 3$$
, then
The equation has two roots.
or
$$w_{1} = \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
Simplify
$$w_{2} = - \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
Simplify
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 4 = 0$$
transform
$$\tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 \cot{\left(x \right)} + 3 = 0$$
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
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
$$a = 1$$
$$b = 3$$
$$c = \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + 3$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (3 + tan(x)^2 + 3*tan(x)) = -3 - 12*tan(x) - 4*tan(x)^2
The equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
Simplify
$$w_{2} = - \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
Simplify
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
pi | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 ||
0 - -- + - re|atan|- - -------|| - I*im|atan|- - -------|| + - re|atan|- + -------|| - I*im|atan|- + -------||
4 \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) - \left(\frac{\pi}{4} + \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || pi | |1 I*\/ 3 || | |1 I*\/ 3 ||
- re|atan|- + -------|| - re|atan|- - -------|| - -- - I*im|atan|- + -------|| - I*im|atan|- - -------||
\ \2 2 // \ \2 2 // 4 \ \2 2 // \ \2 2 //
$$- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - \frac{\pi}{4} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -pi | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| 1*----*|- re|atan|- - -------|| - I*im|atan|- - -------|||*|- re|atan|- + -------|| - I*im|atan|- + -------||| 4 \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$1 \left(- \frac{\pi}{4}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 |||
-pi*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------|||
\ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
--------------------------------------------------------------------------------------------------------
4
$$- \frac{\pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)}{4}$$
-pi*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))/4
Rapid solution
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-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = - re|atan|- - -------|| - I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x3 = - re|atan|- + -------|| - I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
Numerical answer
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x1 = -41.6261027538328
x2 = -91.8915846481544
x3 = -54.192473039953
x4 = -10.2101758562123
x5 = 68.3296401581437
x6 = -54.1924731017967
x7 = -3.92699077227342
x8 = 93.4623811044448
x9 = -47.9092883924126
x10 = -3.92699077276863
x11 = -76.1836216304923
x12 = -85.6083999205894
x13 = 2.35619441168939
x14 = -32.2013244485502
x15 = 84.0376034236404
x16 = -25.918139886171
x17 = 55.7632699272178
x18 = -98.1747706322284
x19 = -19.6349541709315
x20 = -76.1836218812413
x21 = 18.0641575857363
x22 = 11.7809727371379
x23 = 14.9225649923453
x24 = 99.7455671224518
x25 = -0.785398151967561
x26 = 24.3473429940575
x27 = 30.6305285608463
x28 = -98.1747702202347
x29 = 96.6039735663169
x30 = 40.0553062162188
x31 = 46.3384915761999
x32 = 33.7721213315842
x33 = -25.9181399071463
x34 = 74.6128249560098
x35 = -63.6172513370425
x36 = 77.7544185241376
x37 = -3.92699110792993
x38 = 90.3207887399165
x39 = 62.0464548249302
x39 = 62.0464548249302
The graph