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- Equation:
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Equation 51=4x-x
- Equation 2cos^2(2x)-2sin(4x)+1=0
- Equation (x^2-16)^2+(x^2+5x-36)^2=0
- Equation 5sin^2*x-12sinx+4=0
- Express {x} in terms of y where:
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- 5*x+5*y=20
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Identical expressions
- 5sin^ two *x-12sinx+ four = zero
- 5 sinus of squared multiply by x minus 12 sinus of x plus 4 equally 0
- 5 sinus of to the power of two multiply by x minus 12 sinus of x plus four equally zero
- 5sin2*x-12sinx+4=0
- 5sin²*x-12sinx+4=0
- 5sin to the power of 2*x-12sinx+4=0
- 5sin^2x-12sinx+4=0
- 5sin2x-12sinx+4=0
- 5sin^2*x-12sinx+4=O
-
Similar expressions
- 5sin^2*x+12sinx+4=0
- 5sin^2*x-12sinx-4=0
5sin^2*x-12sinx+4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 5*sin (x) - 12*sin(x) + 4 = 0
$$\left(5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)}\right) + 4 = 0$$
Detail solution
Given the equation
$$\left(5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)}\right) + 4 = 0$$
transform
$$5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)} + 4 = 0$$
$$\left(5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)}\right) + 4 = 0$$
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
$$a = 5$$
$$b = -12$$
$$c = 4$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 2$$
$$w_{2} = \frac{2}{5}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$\left(5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)}\right) + 4 = 0$$
transform
$$5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)} + 4 = 0$$
$$\left(5 \sin^{2}{\left(x \right)} - 12 \sin{\left(x \right)}\right) + 4 = 0$$
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
$$a = 5$$
$$b = -12$$
$$c = 4$$
, then
D = b^2 - 4 * a * c =
(-12)^2 - 4 * (5) * (4) = 64
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 2$$
$$w_{2} = \frac{2}{5}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
Rapid solution
[src]
x1 = pi - asin(2/5)
$$x_{1} = \pi - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
x2 = asin(2/5)
$$x_{2} = \operatorname{asin}{\left(\frac{2}{5} \right)}$$
x3 = pi - re(asin(2)) - I*im(asin(2))
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = I*im(asin(2)) + re(asin(2))
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = re(asin(2)) + i*im(asin(2))
Sum and product of roots
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sum
pi - asin(2/5) + asin(2/5) + pi - re(asin(2)) - I*im(asin(2)) + I*im(asin(2)) + re(asin(2))
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) + \left(\left(\operatorname{asin}{\left(\frac{2}{5} \right)} + \left(\pi - \operatorname{asin}{\left(\frac{2}{5} \right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
product
(pi - asin(2/5))*asin(2/5)*(pi - re(asin(2)) - I*im(asin(2)))*(I*im(asin(2)) + re(asin(2)))
$$\left(\pi - \operatorname{asin}{\left(\frac{2}{5} \right)}\right) \operatorname{asin}{\left(\frac{2}{5} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$
=
-(pi - asin(2/5))*(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))*asin(2/5)
$$- \left(\pi - \operatorname{asin}{\left(\frac{2}{5} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \operatorname{asin}{\left(\frac{2}{5} \right)}$$
-(pi - asin(2/5))*(i*im(asin(2)) + re(asin(2)))*(-pi + i*im(asin(2)) + re(asin(2)))*asin(2/5)
Numerical answer
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x1 = -31.0044096898304
x2 = -62.4203362257284
x3 = 50.6769993035042
x4 = 88.3761111465817
x5 = 21.5796317290611
x6 = -68.703521532908
x7 = 82.0929258394021
x8 = 71.8451141864978
x9 = 100.942481760941
x10 = -5.8716684611121
x11 = 25.5442580747858
x12 = -24.7212243826509
x13 = 1391.31402869421
x14 = -60.1017772642736
x15 = -53.818591957094
x16 = -49.8539656113692
x17 = -85.2345184929919
x18 = -43.5707803041896
x19 = 38.110628689145
x20 = 9.01326111470189
x21 = 19.2610727676062
x22 = 2.73007580752231
x23 = -34.9690360355552
x24 = 84.4114848008569
x25 = -93.8362627616263
x26 = -78.9513331858123
x27 = 0.411516846067488
x28 = 6.69470215324707
x29 = -97.8008891073511
x30 = 15.2964464218815
x31 = -16.1194801140165
x32 = 52.995558264959
x33 = -37.28759499701
x34 = -12.1548537682917
x35 = -41.2522213427348
x36 = 59.2787435721386
x37 = 78.1282994936773
x38 = -200.650412983679
x39 = -104.084074414531
x40 = -74.9867068400875
x41 = -3.55310949965728
x42 = -72.6681478786327
x43 = 34.1460023434202
x44 = 96.9778554152161
x45 = 46.7123729577794
x46 = -9.83629480683687
x47 = 31.8274433819654
x48 = 44.3938139963246
x49 = 27.8628170362407
x50 = -22.402665421196
x51 = -47.5354066499144
x52 = -87.5530774544467
x53 = -81.2698921472671
x54 = -118.969003990345
x55 = 40.4291876505998
x56 = 75.8097405322225
x57 = 65.5619288793182
x58 = 63.2433699178634
x59 = -100.119448068806
x60 = 69.5265552250429
x61 = -250.915895441116
x62 = 12.9778874604267
x63 = 94.6592964537613
x64 = -28.6858507283756
x65 = -66.3849625714531
x66 = -91.5177038001715
x67 = -112.685818683165
x68 = -18.4380390754713
x69 = 90.6946701080365
x70 = 56.9601846106838
x71 = -56.1371509185488
x71 = -56.1371509185488