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Equation cos9(x)-cos7(x)=√2sin(x)
Equation cos(2x)+sin((pi/2)-x)+1=0
Equation 2cos^2x+5sinx=0
Equation x^3=125
Express {x} in terms of y where:
sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
(x^2+y^2-1)^3+x^2*y^3=0
-1*x-19*y=-10
4*x-16*y=6
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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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     2                  
2*cos (x) + 5*sin(x) = 0

5 \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0

Simplify

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*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)

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

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)}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

         /    /                         ____________\\         /    /                         ____________\\
         |    |        ____     ____   /       ____ ||         |    |        ____     ____   /       ____ ||
         |    |  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 [src]

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

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Identical expressions

Similar expressions

2cos^2x+5sinx=0 equation

Numerical solution:

The solution