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-
Similar expressions
- cos²(3*x+5pi/4)=1
cos²(3*x-5pi/4)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2/ 5*pi\
cos |3*x - ----| = 1
\ 4 /
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} = 1$$
Detail solution
Given the equation
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} = 1$$
transform
$$\frac{\sin{\left(6 x \right)}}{2} - \frac{1}{2} = 0$$
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} - 1 = 0$$
Do replacement
$$w = \sin{\left(3 x + \frac{\pi}{4} \right)}$$
Given the equation
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} - 1 = 0$$
Because equation degree is equal to = 2 - contains the even number 2 in the numerator, then
the equation has two real roots.
Get the root 2-th degree of the equation sides:
We get:
$$\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \sqrt{1}$$
$$\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \left(-1\right) \sqrt{1}$$
or
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = 1$$
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = -1$$
Expand brackets in the left part
This equation has no roots
Expand brackets in the left part
This equation has no roots
or
do backward replacement
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = w$$
Given the equation
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)} - \frac{\pi}{4}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$3$$
substitute w:
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} = 1$$
transform
$$\frac{\sin{\left(6 x \right)}}{2} - \frac{1}{2} = 0$$
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} - 1 = 0$$
Do replacement
$$w = \sin{\left(3 x + \frac{\pi}{4} \right)}$$
Given the equation
$$\cos^{2}{\left(3 x - \frac{5 \pi}{4} \right)} - 1 = 0$$
Because equation degree is equal to = 2 - contains the even number 2 in the numerator, then
the equation has two real roots.
Get the root 2-th degree of the equation sides:
We get:
$$\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \sqrt{1}$$
$$\sqrt{\left(0 w + \sin{\left(3 x + \frac{\pi}{4} \right)}\right)^{2}} = \left(-1\right) \sqrt{1}$$
or
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = 1$$
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = -1$$
Expand brackets in the left part
sin3*x+pi/4 = 1
This equation has no roots
Expand brackets in the left part
sin3*x+pi/4 = -1
This equation has no roots
or
do backward replacement
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = w$$
Given the equation
$$\sin{\left(3 x + \frac{\pi}{4} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)} - \frac{\pi}{4}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$3$$
substitute w:
Rapid solution
[src]
-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
pi
x2 = --
12
$$x_{2} = \frac{\pi}{12}$$
5*pi
x3 = ----
12
$$x_{3} = \frac{5 \pi}{12}$$
x3 = 5*pi/12
Sum and product of roots
[src]
sum
pi pi 5*pi - -- + -- + ---- 4 12 12
$$\left(- \frac{\pi}{4} + \frac{\pi}{12}\right) + \frac{5 \pi}{12}$$
=
pi -- 4
$$\frac{\pi}{4}$$
product
-pi pi 5*pi ----*--*---- 4 12 12
$$\frac{5 \pi}{12} \cdot - \frac{\pi}{4} \frac{\pi}{12}$$
=
3 -5*pi ------ 576
$$- \frac{5 \pi^{3}}{576}$$
-5*pi^3/576
Numerical answer
[src]
x1 = 59.9520598481687
x2 = 22.2529479607571
x3 = 94.5095789779178
x4 = -41.62610272373
x5 = 79.8488133554185
x6 = 84.0376034179138
x7 = -98.174770348449
x8 = -93.9859802388653
x9 = 40.05530620807
x10 = 50.5272818759395
x11 = -76.1836217744616
x12 = 42.149701543678
x13 = 88.2263937072014
x14 = -1.8325956303714
x15 = 37.9609112670449
x16 = -30.1069295278938
x17 = -43.7204977825881
x18 = 6.5449848141863
x19 = -78.2780169853537
x20 = -32.2013246284748
x21 = 20.1585529574804
x22 = -100.269165531996
x23 = -15.4461636998964
x24 = -85.6083998783531
x25 = -89.797189973996
x26 = 13.8753676266739
x27 = -50.0036830653347
x28 = -8.11578095040642
x29 = -10.2101760567131
x30 = -96.0803752628436
x31 = -54.1924732011039
x32 = 18.0641575904069
x33 = 24.3473430106795
x34 = -17.5405590488288
x35 = 0.261799379333194
x36 = 28.5361333386207
x37 = -67.8060413893714
x38 = -52.0980781057903
x39 = 90.3207887476935
x40 = -48.9564853809923
x41 = 4.45058952366422
x42 = 130.114295757992
x43 = -92.9387829749902
x44 = -37.4373122668668
x45 = 77.7544182162793
x46 = -12.3045714094311
x47 = -21.7293492017758
x48 = -71.9948316512416
x49 = 48.4328866713396
x50 = -6.02138589730989
x51 = 62.0464548168765
x52 = 44.2440965425203
x53 = 2.35619443223801
x54 = -3.92699098017885
x55 = -83.5140047581305
x56 = -87.7027949433722
x57 = 26.4417380971056
x58 = 68.3296401684154
x59 = 13.8753675471005
x60 = -45.814892803953
x61 = 92.41518382174
x62 = 99.7455667784648
x63 = -74.0892266841028
x64 = -61.5228561900619
x65 = -65.7116463631172
x66 = 66.2352451246547
x67 = -23.8237442176309
x68 = -56.2868684488565
x69 = 57.8576647795972
x70 = -39.531707620165
x71 = 46.3384915894085
x72 = 9.68657715106654
x73 = -19.6349541459037
x74 = -58.3812632463393
x75 = 72.5184304232893
x76 = 75.6600230667895
x77 = 33.7721210833439
x78 = -63.6172513012213
x79 = 11.7809725139856
x80 = 47.3856893488309
x81 = 70.424035246248
x82 = 15.9697626852522
x83 = 55.7632696509651
x84 = 35.8665162033444
x85 = -33.2485222538391
x86 = -34.2957199248261
x87 = 64.1408501103489
x88 = 81.9432084286987
x89 = -28.012534480777
x89 = -28.012534480777