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sin^24π₯βsin4π₯β2=0 equation
The teacher will be very surprised to see your correct solution π
The solution
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[src]
2 sin (4*x) - sin(4*x) - 2 = 0
$$\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)} - 2 = 0$$
Detail solution
Given the equation
$$\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)} - 2 = 0$$
transform
$$- \sin{\left(4 x \right)} - \frac{\cos{\left(8 x \right)}}{2} - \frac{3}{2} = 0$$
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)} - 2\right) + 0 = 0$$
Do replacement
$$w = \sin{\left(4 x \right)}$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 2$$
Simplify
$$w_{2} = -1$$
Simplify
do backward replacement
$$\sin{\left(4 x \right)} = w$$
Given the equation
$$\sin{\left(4 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$4$$
substitute w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{2} \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(-1 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{4} + \frac{\pi}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{2} \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(-1 \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} + \frac{3 \pi}{8}$$
$$\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)} - 2 = 0$$
transform
$$- \sin{\left(4 x \right)} - \frac{\cos{\left(8 x \right)}}{2} - \frac{3}{2} = 0$$
$$\left(\sin^{2}{\left(4 x \right)} - \sin{\left(4 x \right)} - 2\right) + 0 = 0$$
Do replacement
$$w = \sin{\left(4 x \right)}$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-2) = 9
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 2$$
Simplify
$$w_{2} = -1$$
Simplify
do backward replacement
$$\sin{\left(4 x \right)} = w$$
Given the equation
$$\sin{\left(4 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$4 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$4$$
substitute w:
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(w_{2} \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\operatorname{asin}{\left(-1 \right)}}{4}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{3} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{4} + \frac{\pi}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{3} = \frac{\pi n}{2} + \frac{\pi}{4} - \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(w_{2} \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} - \frac{\operatorname{asin}{\left(-1 \right)}}{4} + \frac{\pi}{4}$$
$$x_{4} = \frac{\pi n}{2} + \frac{3 \pi}{8}$$
Rapid solution
[src]
-pi
x1 = ----
8
$$x_{1} = - \frac{\pi}{8}$$
3*pi
x2 = ----
8
$$x_{2} = \frac{3 \pi}{8}$$
re(asin(2)) pi I*im(asin(2))
x3 = - ----------- + -- - -------------
4 4 4
$$x_{3} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{\pi}{4} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}$$
re(asin(2)) I*im(asin(2))
x4 = ----------- + -------------
4 4
$$x_{4} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}$$
Sum and product of roots
[src]
sum
pi 3*pi re(asin(2)) pi I*im(asin(2)) re(asin(2)) I*im(asin(2))
0 - -- + ---- + - ----------- + -- - ------------- + ----------- + -------------
8 8 4 4 4 4 4
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right) - \left(- \frac{\pi}{2} + \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right)$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
-pi 3*pi / re(asin(2)) pi I*im(asin(2))\ /re(asin(2)) I*im(asin(2))\ 1*----*----*|- ----------- + -- - -------------|*|----------- + -------------| 8 8 \ 4 4 4 / \ 4 4 /
$$\frac{3 \pi}{8} \cdot 1 \left(- \frac{\pi}{8}\right) \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{\pi}{4} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right) \left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}}{4}\right)$$
=
2
3*pi *(I*im(asin(2)) + re(asin(2)))*(-pi + I*im(asin(2)) + re(asin(2)))
-----------------------------------------------------------------------
1024
$$\frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)}{1024}$$
3*pi^2*(i*im(asin(2)) + re(asin(2)))*(-pi + i*im(asin(2)) + re(asin(2)))/1024
Numerical answer
[src]
x1 = -6.67588458616306
x2 = -556.45459894507
x3 = 42.0188017149397
x4 = -744.950158056443
x5 = -47.5165888974914
x6 = -45.9457927193195
x7 = -25.525440336424
x8 = 100.138265889851
x9 = -53.7997741488835
x10 = -82.0741080211412
x11 = -17.6714586996829
x12 = -60.0829594427331
x13 = 4.31968985730964
x14 = 20.0276531232969
x15 = -17.6714587216416
x16 = 62.4391539349309
x17 = -30.2378292874754
x18 = 56.1559687796345
x19 = 71.8639320174744
x20 = 70.2931355961025
x21 = 79.7179134126456
x22 = 35.7356163522263
x23 = -1.96349557016575
x24 = 12.1736716073441
x25 = -97.7820713164945
x26 = -8.24668073366495
x27 = 302.770992033397
x28 = -55.3705706599853
x29 = 5.89048628952707
x30 = -9.81747697874728
x31 = 48.3019870161865
x32 = -39.6626073002566
x33 = 92.2842841763354
x34 = 101.709061988899
x35 = 93.8550805924588
x36 = 34.1648201938636
x37 = 78.1471173590006
x38 = 64.0099503038507
x39 = -31.8086255641596
x40 = -91.4988860046927
x41 = -75.7909227329734
x42 = 43.5895979321204
x43 = -89.9280898521286
x44 = -96.2112749750547
x45 = 84.4303025069203
x46 = -38.091810864829
x47 = 86.0010988906877
x48 = 32.5940236663725
x49 = -1.96349547204618
x50 = -83.6449044564232
x51 = 57.7267648828298
x52 = 26.3108384365878
x53 = -23.954644072489
x54 = 98.567469621515
x55 = 49.8727834419781
x56 = -16.1006622874121
x57 = -85.215700846439
x58 = -3.53429177195145
x59 = 18.4568567951268
x60 = -45.9457926797218
x61 = 27.8816348659918
x62 = 13.7444678134058
x63 = 51.4435795572927
x64 = -74.2201264093036
x65 = -67.9369412926542
x66 = -69.5077374540623
x67 = -52.2289778464515
x68 = -61.6537558785227
x69 = 40.448005364229
x69 = 40.448005364229
The graph