Mister Exam
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Similar expressions
- (cos2x)/(sqrt(2)cosx-sinx)=0
(cos2x)/(sqrt(2)cosx+sinx)=0 equation
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The solution
You have entered
[src]
cos(2*x) --------------------- = 0 ___ \/ 2 *cos(x) + sin(x)
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
Detail solution
Given the equation
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
transform
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$\frac{\cos{\left(2 x \right)}}{\sqrt{2} w + \sin{\left(x \right)}} = 0$$
Multiply the equation sides by the denominator w*sqrt(2) + sin(x)
we get:
$$\cos{\left(2 x \right)} = 0$$
Expand brackets in the left part
This equation has no roots
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
transform
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
$$\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \sqrt{2} \cos{\left(x \right)}} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$\frac{\cos{\left(2 x \right)}}{\sqrt{2} w + \sin{\left(x \right)}} = 0$$
Multiply the equation sides by the denominator w*sqrt(2) + sin(x)
we get:
$$\cos{\left(2 x \right)} = 0$$
Expand brackets in the left part
cos2*x = 0
This equation has no roots
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
Sum and product of roots
[src]
sum
pi 3*pi -- + ---- 4 4
$$\frac{\pi}{4} + \frac{3 \pi}{4}$$
=
pi
$$\pi$$
product
pi 3*pi --*---- 4 4
$$\frac{\pi}{4} \frac{3 \pi}{4}$$
=
2 3*pi ----- 16
$$\frac{3 \pi^{2}}{16}$$
3*pi^2/16
Rapid solution
[src]
pi
x1 = --
4
$$x_{1} = \frac{\pi}{4}$$
3*pi
x2 = ----
4
$$x_{2} = \frac{3 \pi}{4}$$
x2 = 3*pi/4
Numerical answer
[src]
x1 = 32.2013246992954
x2 = 66.7588438887831
x3 = -27.4889357189107
x4 = 10.2101761241668
x5 = 44.7676953136546
x6 = -21.2057504117311
x7 = 79.3252145031423
x8 = -52.621676947629
x9 = -3.92699081698724
x10 = 82.4668071567321
x11 = -69.9004365423729
x12 = 29.0597320457056
x13 = -58.9048622548086
x14 = 95.0331777710912
x15 = -77.7544181763474
x16 = -14.9225651045515
x17 = 88.7499924639117
x18 = 16.4933614313464
x19 = 41.6261026600648
x20 = 96.6039740978861
x21 = 0.785398163397448
x22 = -91.8915851175014
x23 = 73.0420291959627
x24 = -74.6128255227576
x25 = -46.3384916404494
x26 = -5.49778714378214
x27 = 84.037603483527
x28 = -84.037603483527
x29 = 91.8915851175014
x30 = -30.6305283725005
x31 = -55.7632696012188
x32 = -71.4712328691678
x33 = -36.9137136796801
x34 = -65.1880475619882
x35 = 60.4756585816035
x36 = 3.92699081698724
x37 = 38.484510006475
x38 = 90.3207887907066
x39 = -90.3207887907066
x40 = 98.174770424681
x41 = -18.0641577581413
x42 = 76.1836218495525
x43 = 24.3473430653209
x44 = -80.8960108299372
x45 = -40.0553063332699
x46 = 22.776546738526
x47 = -68.329640215578
x48 = 35.3429173528852
x49 = 51.0508806208341
x50 = 46.3384916404494
x51 = 40.0553063332699
x52 = -33.7721210260903
x53 = -25.9181393921158
x54 = -96.6039740978861
x55 = -49.4800842940392
x56 = -87.1791961371168
x57 = 25.9181393921158
x58 = 85.6083998103219
x59 = -99.7455667514759
x60 = 13.3517687777566
x61 = -8.63937979737193
x62 = 62.0464549083984
x63 = 54.1924732744239
x64 = -24.3473430653209
x65 = 68.329640215578
x66 = 63.6172512351933
x67 = 19.6349540849362
x68 = 18.0641577581413
x69 = -93.4623814442964
x70 = -43.1968989868597
x71 = -11.7809724509617
x72 = -62.0464549083984
x73 = 69.9004365423729
x74 = 57.3340659280137
x75 = -63.6172512351933
x76 = -1808.77197030432
x77 = -2.35619449019234
x78 = 47.9092879672443
x79 = -47.9092879672443
x80 = 7.06858347057703
x80 = 7.06858347057703