Mister Exam
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How to use it?
- Equation:
-
Equation 14/(x-3)=-7/9
-
Equation x^4+2*x^2-8=0
-
Equation 9x−9(x+1)=−2x
-
Equation x^4+4=0
- Express {x} in terms of y where:
- -17*x+1*y=-11
- 1*x+4*y=20
- -18*x-2*y=9
- -20*x-14*y=5
-
Identical expressions
- one - two *cos(3x)+ five *cos(3x)+ one = zero
- 1 minus 2 multiply by co sinus of e of (3x) plus 5 multiply by co sinus of e of (3x) plus 1 equally 0
- one minus two multiply by co sinus of e of (3x) plus five multiply by co sinus of e of (3x) plus one equally zero
- 1-2cos(3x)+5cos(3x)+1=0
- 1-2cos3x+5cos3x+1=0
- 1-2*cos(3x)+5*cos(3x)+1=O
-
Similar expressions
- 1-2*cos(3x)-5*cos(3x)+1=0
- 1+2*cos(3x)+5*cos(3x)+1=0
- 1-2*cos(3x)+5*cos(3x)-1=0
1-2*cos(3x)+5*cos(3x)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
1 - 2*cos(3*x) + 5*cos(3*x) + 1 = 0
$$\left(\left(1 - 2 \cos{\left(3 x \right)}\right) + 5 \cos{\left(3 x \right)}\right) + 1 = 0$$
Detail solution
Given the equation
$$\left(\left(1 - 2 \cos{\left(3 x \right)}\right) + 5 \cos{\left(3 x \right)}\right) + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$3 \cos{\left(3 x \right)} = -2$$
The equation is transformed to
$$\cos{\left(3 x \right)} = - \frac{2}{3}$$
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
Or
$$3 x = \pi n + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{3} + \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
$$\left(\left(1 - 2 \cos{\left(3 x \right)}\right) + 5 \cos{\left(3 x \right)}\right) + 1 = 0$$
- this is the simplest trigonometric equation
Move 2 to right part of the equation
with the change of sign in 2
We get:
$$3 \cos{\left(3 x \right)} = -2$$
Divide both parts of the equation by 3
The equation is transformed to
$$\cos{\left(3 x \right)} = - \frac{2}{3}$$
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
Or
$$3 x = \pi n + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{2}{3} \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{3} + \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
Rapid solution
[src]
acos(-2/3) 2*pi
x1 = - ---------- + ----
3 3
$$x_{1} = - \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3} + \frac{2 \pi}{3}$$
acos(-2/3)
x2 = ----------
3
$$x_{2} = \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
x2 = acos(-2/3)/3
Sum and product of roots
[src]
sum
acos(-2/3) 2*pi acos(-2/3)
- ---------- + ---- + ----------
3 3 3
$$\frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3} + \left(- \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3} + \frac{2 \pi}{3}\right)$$
=
2*pi ---- 3
$$\frac{2 \pi}{3}$$
product
/ acos(-2/3) 2*pi\ acos(-2/3) |- ---------- + ----|*---------- \ 3 3 / 3
$$\left(- \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3} + \frac{2 \pi}{3}\right) \frac{\operatorname{acos}{\left(- \frac{2}{3} \right)}}{3}$$
=
(-acos(-2/3) + 2*pi)*acos(-2/3)
-------------------------------
9
$$\frac{\left(- \operatorname{acos}{\left(- \frac{2}{3} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \frac{2}{3} \right)}}{9}$$
(-acos(-2/3) + 2*pi)*acos(-2/3)/9
Numerical answer
[src]
x1 = 22.2715047986512
x2 = 48.9379286827174
x3 = 4.95563153246035
x4 = -7.61073908189883
x5 = -15.9883194914716
x6 = -5.51634397950563
x7 = -91.3865431776266
x8 = 1.32755377471924
x9 = -1.32755377471924
x10 = 57.8762215393355
x11 = -82.4482503210086
x12 = -39.0266656177968
x13 = -78.2594601162222
x14 = 42.6547433755379
x15 = -99.7641235871994
x16 = -59.9706166417287
x17 = 40.5603482731447
x18 = -377.757959758449
x19 = 95.575333382413
x20 = 71.9762748090426
x21 = -45.3098509249763
x22 = 15.9883194914716
x23 = 99.7641235871994
x24 = -27.9939776587855
x25 = 78.2594601162222
x26 = 27.9939776587855
x27 = 25.8995825563923
x28 = -67.7874846042562
x29 = 69.8818797066494
x30 = -32.1827678635719
x31 = -62.0650117441219
x32 = -101.858518689593
x33 = 86.637040525795
x34 = 20.177109696258
x35 = -36.3715580683583
x36 = 76.165065013829
x37 = -13.8939243890784
x38 = 45.3098509249763
x39 = -65.693089501863
x40 = 64.1594068465151
x41 = -9.70513418429202
x42 = -18.0827145938648
x43 = 74.0706699114358
x44 = 97.6697284848062
x45 = 68.3481970513015
x46 = -23.8051874539991
x47 = 59.4099041946834
x48 = -34.2771629659651
x49 = -49.4986411297627
x50 = 66.2538019489083
x51 = -57.8762215393355
x52 = 34.2771629659651
x53 = 38.4659531707515
x54 = 82.4482503210086
x55 = 59.9706166417287
x56 = -93.4809382800198
x57 = 92.9202258329746
x58 = 30.0883727611787
x59 = 53.6874313345491
x60 = -25.8995825563923
x61 = 32.1827678635719
x62 = 36.3715580683583
x63 = -69.8818797066494
x64 = -51.5930362321559
x65 = 7.61073908189883
x66 = -95.575333382413
x67 = -80.3538552186154
x68 = -21.7107923516059
x69 = -53.6874313345491
x70 = 89.2921480752335
x71 = -3.42194887711244
x72 = 24.3658999010444
x73 = -86.637040525795
x74 = 13.8939243890784
x75 = 11.7995292866852
x76 = 84.5426454234018
x77 = -71.9762748090426
x78 = 62.0650117441219
x79 = -76.165065013829
x80 = 18.0827145938648
x81 = -97.6697284848062
x82 = -19.6163972492127
x83 = -111.769781754513
x84 = 55.7818264369423
x85 = -47.4042460273695
x86 = 51.5930362321559
x87 = -74.0706699114358
x88 = -11.7995292866852
x89 = -55.7818264369423
x90 = 9.70513418429202
x91 = 80.3538552186154
x92 = -30.0883727611787
x92 = -30.0883727611787