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6*cos^2(x)-13*sin(x)-13=0 equation
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The solution
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[src]
2 6*cos (x) - 13*sin(x) - 13 = 0
$$\left(- 13 \sin{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) - 13 = 0$$
Detail solution
Given the equation
$$\left(- 13 \sin{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) - 13 = 0$$
transform
$$- 6 \sin^{2}{\left(x \right)} - 13 \sin{\left(x \right)} - 7 = 0$$
$$- 6 \sin^{2}{\left(x \right)} - 13 \sin{\left(x \right)} - 7 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = -6$$
$$b = -13$$
$$c = -7$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{7}{6}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{7}{6} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{7}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{7}{6} \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{7}{6} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
$$\left(- 13 \sin{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) - 13 = 0$$
transform
$$- 6 \sin^{2}{\left(x \right)} - 13 \sin{\left(x \right)} - 7 = 0$$
$$- 6 \sin^{2}{\left(x \right)} - 13 \sin{\left(x \right)} - 7 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = -6$$
$$b = -13$$
$$c = -7$$
, then
D = b^2 - 4 * a * c =
(-13)^2 - 4 * (-6) * (-7) = 1
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{7}{6}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{7}{6} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{7}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{7}{6} \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{7}{6} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Sum and product of roots
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sum
/ / ____\\ / / ____\\ / / ____\\ / / ____\\ pi | |6 I*\/ 13 || | |6 I*\/ 13 || | |6 I*\/ 13 || | |6 I*\/ 13 || - -- + - 2*re|atan|- - --------|| - 2*I*im|atan|- - --------|| + - 2*re|atan|- + --------|| - 2*I*im|atan|- + --------|| 2 \ \7 7 // \ \7 7 // \ \7 7 // \ \7 7 //
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)}\right) + \left(- \frac{\pi}{2} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)}\right)\right)$$
=
/ / ____\\ / / ____\\ / / ____\\ / / ____\\
| |6 I*\/ 13 || | |6 I*\/ 13 || pi | |6 I*\/ 13 || | |6 I*\/ 13 ||
- 2*re|atan|- - --------|| - 2*re|atan|- + --------|| - -- - 2*I*im|atan|- - --------|| - 2*I*im|atan|- + --------||
\ \7 7 // \ \7 7 // 2 \ \7 7 // \ \7 7 //
$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} - \frac{\pi}{2} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)}$$
product
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ -pi | | |6 I*\/ 13 || | |6 I*\/ 13 ||| | | |6 I*\/ 13 || | |6 I*\/ 13 ||| ----*|- 2*re|atan|- - --------|| - 2*I*im|atan|- - --------|||*|- 2*re|atan|- + --------|| - 2*I*im|atan|- + --------||| 2 \ \ \7 7 // \ \7 7 /// \ \ \7 7 // \ \7 7 ///
$$- \frac{\pi}{2} \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)}\right)$$
=
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\
| | |6 I*\/ 13 || | |6 I*\/ 13 ||| | | |6 I*\/ 13 || | |6 I*\/ 13 |||
-2*pi*|I*im|atan|- - --------|| + re|atan|- - --------|||*|I*im|atan|- + --------|| + re|atan|- + --------|||
\ \ \7 7 // \ \7 7 /// \ \ \7 7 // \ \7 7 ///
$$- 2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)}\right)$$
-2*pi*(i*im(atan(6/7 - i*sqrt(13)/7)) + re(atan(6/7 - i*sqrt(13)/7)))*(i*im(atan(6/7 + i*sqrt(13)/7)) + re(atan(6/7 + i*sqrt(13)/7)))
Rapid solution
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-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
/ / ____\\ / / ____\\
| |6 I*\/ 13 || | |6 I*\/ 13 ||
x2 = - 2*re|atan|- - --------|| - 2*I*im|atan|- - --------||
\ \7 7 // \ \7 7 //
$$x_{2} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} - \frac{\sqrt{13} i}{7} \right)}\right)}$$
/ / ____\\ / / ____\\
| |6 I*\/ 13 || | |6 I*\/ 13 ||
x3 = - 2*re|atan|- + --------|| - 2*I*im|atan|- + --------||
\ \7 7 // \ \7 7 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{6}{7} + \frac{\sqrt{13} i}{7} \right)}\right)}$$
x3 = -2*re(atan(6/7 + sqrt(13)*i/7)) - 2*i*im(atan(6/7 + sqrt(13)*i/7))
Numerical answer
[src]
x1 = -70.6858337669459
x2 = 80.1106131052551
x3 = -20.4203517979426
x4 = -89.5353907815952
x5 = -39.2699086294832
x6 = -64.4026498150031
x7 = -76.9690210377331
x8 = -26.7035385231271
x9 = -7.8539814796249
x10 = 10.9955733409181
x11 = 86.3937978748502
x12 = 98.9601698149885
x13 = -14.1371668165022
x14 = 98.9601677166653
x15 = 48.694685722013
x16 = 36.128315762565
x17 = 29.8451303493858
x18 = 4.71238854668104
x19 = -58.1194639820442
x20 = -83.2522068648451
x21 = 92.6769828965169
x22 = 54.9778705291622
x23 = 67.5442425570876
x24 = -70.6858351975079
x25 = -83.2522058064823
x26 = -26.703536577867
x27 = -1.57079645232832
x28 = 42.4115007114988
x29 = 61.2610578024167
x30 = -64.4026489742146
x31 = 17.2787606119261
x32 = -45.553093616679
x33 = -51.8362786849095
x34 = -95.8185757995315
x35 = 61.2610551520466
x36 = 23.5619453791571
x37 = -32.9867238480523
x38 = -95.8185758671642
x39 = 73.8274275157047
x39 = 73.8274275157047