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Similar expressions
- 2cos^2x-5sinx=0
2cos^2x+5sinx=0 equation
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The solution
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[src]
2 2*cos (x) + 5*sin(x) = 0
$$5 \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
Detail solution
Given the equation
$$5 \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
transform
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 2 = 0$$
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
$$a = -2$$
$$b = 5$$
$$c = 2$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{5}{4} - \frac{\sqrt{41}}{4}$$
$$w_{2} = \frac{5}{4} + \frac{\sqrt{41}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$5 \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
transform
$$5 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 2 = 0$$
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
$$a = -2$$
$$b = 5$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (-2) * (2) = 41
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{5}{4} - \frac{\sqrt{41}}{4}$$
$$w_{2} = \frac{5}{4} + \frac{\sqrt{41}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} - \frac{\sqrt{41}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
Rapid solution
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/ / ____________\\ / / ____________\\
| | ____ ____ / ____ || | | ____ ____ / ____ ||
| | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
x1 = 2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------||
\ \ 4 4 4 // \ \ 4 4 4 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}$$
/ ____________\
| ____ ____ / ____ |
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
x2 = -2*atan|- + ------ + ----------------------|
\4 4 4 /
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)}$$
/ / ____________\\ / / ____________\\
| | ____ ____ / ____ || | | ____ ____ / ____ ||
| |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
x3 = - 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------||
\ \4 4 4 // \ \4 4 4 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}$$
/ ____________\
| ____ ____ / ____ |
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
x4 = -2*atan|- + ------ - ----------------------|
\4 4 4 /
$$x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
x4 = -2*atan(-sqrt(10)*sqrt(5 + sqrt(41))/4 + 5/4 + sqrt(41)/4)
Sum and product of roots
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sum
/ / ____________\\ / / ____________\\ / ____________\ / / ____________\\ / / ____________\\ / ____________\
| | ____ ____ / ____ || | | ____ ____ / ____ || | ____ ____ / ____ | | | ____ ____ / ____ || | | ____ ____ / ____ || | ____ ____ / ____ |
| | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || |5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------|| - 2*atan|- + ------ + ----------------------| + - 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|| - 2*atan|- + ------ - ----------------------|
\ \ 4 4 4 // \ \ 4 4 4 // \4 4 4 / \ \4 4 4 // \ \4 4 4 // \4 4 4 /
$$- 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) + \left(- 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right)\right)\right)$$
=
/ ____________\ / ____________\ / / ____________\\ / / ____________\\ / / ____________\\ / / ____________\\
| ____ ____ / ____ | | ____ ____ / ____ | | | ____ ____ / ____ || | | ____ ____ / ____ || | | ____ ____ / ____ || | | ____ ____ / ____ ||
|5 \/ 41 \/ 10 *\/ 5 + \/ 41 | |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||
- 2*atan|- + ------ - ----------------------| - 2*atan|- + ------ + ----------------------| - 2*re|atan|- - ------ + ----------------------|| + 2*re|atan|- - + ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------||
\4 4 4 / \4 4 4 / \ \4 4 4 // \ \ 4 4 4 // \ \4 4 4 // \ \ 4 4 4 //
$$- 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} - 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}$$
product
/ / / ____________\\ / / ____________\\\ / ____________\ / / / ____________\\ / / ____________\\\ / ____________\ | | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ | | | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ | | | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | |2*re|atan|- - + ------ + ----------------------|| + 2*I*im|atan|- - + ------ + ----------------------|||*-2*atan|- + ------ + ----------------------|*|- 2*re|atan|- - ------ + ----------------------|| - 2*I*im|atan|- - ------ + ----------------------|||*-2*atan|- + ------ - ----------------------| \ \ \ 4 4 4 // \ \ 4 4 4 /// \4 4 4 / \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 /
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)}\right)$$
=
/ / / ____________\\ / / ____________\\\ / / / ____________\\ / / ____________\\\ / ____________\ / ____________\
| | | ____ ____ / ____ || | | ____ ____ / ____ ||| | | | ____ ____ / ____ || | | ____ ____ / ____ ||| | ____ ____ / ____ | | ____ ____ / ____ |
| | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | | 5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| | | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 || | |5 \/ 41 \/ 10 *\/ 5 - \/ 41 ||| |5 \/ 41 \/ 10 *\/ 5 + \/ 41 | |5 \/ 41 \/ 10 *\/ 5 + \/ 41 |
-16*|I*im|atan|- - + ------ + ----------------------|| + re|atan|- - + ------ + ----------------------|||*|I*im|atan|- - ------ + ----------------------|| + re|atan|- - ------ + ----------------------|||*atan|- + ------ - ----------------------|*atan|- + ------ + ----------------------|
\ \ \ 4 4 4 // \ \ 4 4 4 /// \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 / \4 4 4 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{41}}{4} + \frac{5}{4} + \frac{\sqrt{10} \sqrt{5 - \sqrt{41}}}{4} \right)}\right)}\right) \operatorname{atan}{\left(\frac{5}{4} + \frac{\sqrt{41}}{4} + \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} \right)} \operatorname{atan}{\left(- \frac{\sqrt{10} \sqrt{5 + \sqrt{41}}}{4} + \frac{5}{4} + \frac{\sqrt{41}}{4} \right)}$$
-16*(i*im(atan(-5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)) + re(atan(-5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)))*(i*im(atan(5/4 - sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)) + re(atan(5/4 - sqrt(41)/4 + sqrt(10)*sqrt(5 - sqrt(41))/4)))*atan(5/4 + sqrt(41)/4 - sqrt(10)*sqrt(5 + sqrt(41))/4)*atan(5/4 + sqrt(41)/4 + sqrt(10)*sqrt(5 + sqrt(41))/4)
Numerical answer
[src]
x1 = 68.7566333480279
x2 = -503.013229605315
x3 = -25.4911462596659
x4 = -6.64159033812716
x5 = -71.8982260016177
x6 = 78.8982213706924
x7 = -333.367226311466
x8 = 66.3318507563332
x9 = 28.6327389132557
x10 = -78.1814113087973
x11 = 62.4734480408483
x12 = 45716.0978900077
x13 = -38.0575168740251
x14 = -82.0398140242822
x15 = -56.9070727955638
x16 = 87.6061892695666
x17 = -97.030967230336
x18 = -34.1991141585402
x19 = -31.7743315668455
x20 = 91.4645919850516
x21 = -2.78318762264222
x22 = 41.1991095276149
x23 = -69.473443409923
x24 = 85.181406677872
x25 = 72.6150360635128
x26 = 18.4911508905912
x27 = 16.0663682988965
x28 = 31.0575215049504
x29 = -27.9159288513606
x30 = 60.0486654491536
x31 = 34.9159242204353
x32 = -84.4645966159768
x33 = -44.3407021812047
x34 = -100.889369945821
x35 = 22.3495536060761
x36 = -63.1902581027434
x37 = -19.2079609524863
x38 = 47.4822948347945
x39 = 49.9070774264891
x40 = 53.7654801419741
x41 = 12.2079655834116
x42 = -53.0486700800789
x43 = -9.06637292982181
x44 = 5.92478027623201
x45 = 3.49999768453737
x46 = -12.9247756453067
x47 = -21.632743544181
x48 = 56.1902627336687
x49 = 9.78318299171695
x50 = -0.358405030947573
x51 = -88.3229993314618
x52 = 24.7743361977708
x53 = -46.7654847728993
x54 = -285.526526445724
x55 = -50.6238874883843
x56 = -94.6061846386414
x57 = -75.7566287171026
x58 = 43.6238921193095
x59 = 81.323003962387
x60 = -197.561932145209
x61 = 97.7477772922312
x62 = -967.968942336604
x63 = -15.3495582370014
x64 = -40.4822994657197
x65 = -65.6150406944381
x66 = -59.3318553872585
x67 = 75.0398186552075
x68 = 100.172559883926
x69 = 93.8893745767462
x70 = 37.3407068121299
x71 = -90.7477819231564
x71 = -90.7477819231564