Mister Exam
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- Equation:
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Equation -x-2+3(x-3)=3(4-x)-3
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Equation 4x=24+x
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Equation x+3=-9x
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Identical expressions
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Similar expressions
- 4cos^2x-sinx-1=0
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4cos^2x-sinx+1=0 equation
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The solution
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[src]
2 4*cos (x) - sin(x) + 1 = 0
$$\left(- \sin{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) + 1 = 0$$
Detail solution
Given the equation
$$\left(- \sin{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) + 1 = 0$$
transform
$$- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 0$$
$$- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 0$$
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
$$a = -4$$
$$b = -1$$
$$c = 5$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{5}{4}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{5}{4} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{5}{4} \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{5}{4} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$\left(- \sin{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) + 1 = 0$$
transform
$$- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 0$$
$$- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 0$$
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
$$a = -4$$
$$b = -1$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (-4) * (5) = 81
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{5}{4}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{5}{4} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{5}{4} \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{5}{4} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
Sum and product of roots
[src]
sum
pi / /4 3*I\\ / /4 3*I\\ / /4 3*I\\ / /4 3*I\\ -- + - 2*re|atan|- - ---|| - 2*I*im|atan|- - ---|| + - 2*re|atan|- + ---|| - 2*I*im|atan|- + ---|| 2 \ \5 5 // \ \5 5 // \ \5 5 // \ \5 5 //
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right) + \left(\frac{\pi}{2} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right)\right)$$
=
pi / /4 3*I\\ / /4 3*I\\ / /4 3*I\\ / /4 3*I\\ -- - 2*re|atan|- - ---|| - 2*re|atan|- + ---|| - 2*I*im|atan|- - ---|| - 2*I*im|atan|- + ---|| 2 \ \5 5 // \ \5 5 // \ \5 5 // \ \5 5 //
$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} + \frac{\pi}{2} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}$$
product
pi / / /4 3*I\\ / /4 3*I\\\ / / /4 3*I\\ / /4 3*I\\\ --*|- 2*re|atan|- - ---|| - 2*I*im|atan|- - ---|||*|- 2*re|atan|- + ---|| - 2*I*im|atan|- + ---||| 2 \ \ \5 5 // \ \5 5 /// \ \ \5 5 // \ \5 5 ///
$$\frac{\pi}{2} \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right)$$
=
/ / /4 3*I\\ / /4 3*I\\\ / / /4 3*I\\ / /4 3*I\\\
2*pi*|I*im|atan|- - ---|| + re|atan|- - ---|||*|I*im|atan|- + ---|| + re|atan|- + ---|||
\ \ \5 5 // \ \5 5 /// \ \ \5 5 // \ \5 5 ///
$$2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right)$$
2*pi*(i*im(atan(4/5 - 3*i/5)) + re(atan(4/5 - 3*i/5)))*(i*im(atan(4/5 + 3*i/5)) + re(atan(4/5 + 3*i/5)))
Rapid solution
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
/ /4 3*I\\ / /4 3*I\\
x2 = - 2*re|atan|- - ---|| - 2*I*im|atan|- - ---||
\ \5 5 // \ \5 5 //
$$x_{2} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}$$
/ /4 3*I\\ / /4 3*I\\
x3 = - 2*re|atan|- + ---|| - 2*I*im|atan|- + ---||
\ \5 5 // \ \5 5 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}$$
x3 = -2*re(atan(4/5 + 3*i/5)) - 2*i*im(atan(4/5 + 3*i/5))
Numerical answer
[src]
x1 = 51.8362789003523
x2 = -29.8451301660968
x3 = -80.1106125794063
x4 = 58.1194644304554
x5 = 1.5707965487099
x6 = 58.1194639689786
x7 = -98.9601688497483
x8 = 32.986723027975
x9 = 26.7035373432502
x10 = 14.1371671054705
x11 = -400.5530636364
x12 = 102.101761119382
x13 = 83.2522051441511
x14 = -23.561945009324
x15 = 95.8185760594783
x16 = 76.9690197502489
x17 = 14.1371668519517
x18 = -73.8274270126378
x19 = 89.53539085929
x20 = 70.6858344990791
x21 = 83.2522055898627
x22 = 32.9867225994861
x23 = -61.2610569677124
x24 = -29.8451300962681
x25 = -42.411500606946
x26 = 7.8539817408679
x27 = -54.9778712408359
x28 = -17.278759767136
x29 = -54.9778717005513
x30 = 11015.994639417
x31 = 45.5530937041236
x32 = 76.9690201583539
x33 = -36.1283154190274
x34 = 39.2699080118312
x35 = -98.9601683784854
x36 = 7.8539816297258
x37 = -48.6946858609913
x38 = 64.4026493085237
x39 = -86.3937977628397
x40 = -4.71238911743622
x41 = -67.544242167931
x42 = -4.71238871026182
x43 = -92.6769830121323
x44 = -17.2787598123494
x45 = -73.8274272800187
x46 = -48.6946862430865
x47 = -10.9955745507927
x48 = 76.9690201096197
x49 = -23.5619452560558
x50 = 39.269908440032
x51 = -92.6769833644259
x52 = 20.4203521494903
x53 = -10.9955741052319
x54 = -17.2787599640228
x54 = -17.2787599640228