Mister Exam
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How to use it?
- Equation:
-
Equation cos(2x)-cos(4x)=sin(6x)
- Equation (15*x-28)/2=76
-
Equation cos(z)=3
-
Equation 4*sin(45-a)*sin(15+a)*cos(15-a)=cos(45-3*a)
- Express {x} in terms of y where:
- 5*x-5*y=20
- 8*x-7*y=-13
- 15*x-15*y=16
- 9*x+18*y=6
- Derivative of:
-
sin(6x)
- Integral of d{x}:
- sin(6x)
-
Identical expressions
- cos(2x)-cos(4x)=sin(6x)
- co sinus of e of (2x) minus co sinus of e of (4x) equally sinus of (6x)
- cos2x-cos4x=sin6x
-
Similar expressions
- sin(6*x)=9/8
- sin6x=1/2
- cos(2x)+cos(4x)=sin(6x)
- sin(6*x)+cos(6*x)*sqrt(3)=-2*cos(8*x)
- sin(6*x-pi/6)=1/2
cos(2x)-cos(4x)=sin(6x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
cos(2*x) - cos(4*x) = sin(6*x)
$$\cos{\left(2 x \right)} - \cos{\left(4 x \right)} = \sin{\left(6 x \right)}$$
Detail solution
Given the equation:
$$\cos{\left(2 x \right)} - \cos{\left(4 x \right)} = \sin{\left(6 x \right)}$$
Transform
$$- \sin{\left(6 x \right)} + \cos{\left(2 x \right)} - \cos{\left(4 x \right)} = 0$$
$$- 4 \sin{\left(3 x \right)} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
Consider each factor separately
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$2 x + \frac{\pi}{4} = 2 \pi n + \operatorname{acos}{\left(0 \right)}$$
$$2 x + \frac{\pi}{4} = 2 \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$2 x + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
$$2 x + \frac{\pi}{4} = 2 \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation with the opposite sign, in total:
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n - \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{8}$$
$$x = \pi n - \frac{3 \pi}{8}$$
$$\sin{\left(3 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(3 x \right)} = 0$$
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$3 x = 2 \pi n$$
$$3 x = 2 \pi n + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{8}$$
$$x = \pi n - \frac{3 \pi}{8}$$
$$x = \frac{2 \pi n}{3}$$
$$x = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
$$\sin{\left(x + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x + \frac{\pi}{4} = 2 \pi n$$
$$x + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation with the opposite sign, in total:
$$x = 2 \pi n - \frac{\pi}{4}$$
$$x = 2 \pi n + \frac{3 \pi}{4}$$
The final answer:
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n - \frac{3 \pi}{8}$$
$$x_{3} = \frac{2 \pi n}{3}$$
$$x_{4} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
$$x_{5} = 2 \pi n - \frac{\pi}{4}$$
$$x_{6} = 2 \pi n + \frac{3 \pi}{4}$$
$$\cos{\left(2 x \right)} - \cos{\left(4 x \right)} = \sin{\left(6 x \right)}$$
Transform
$$- \sin{\left(6 x \right)} + \cos{\left(2 x \right)} - \cos{\left(4 x \right)} = 0$$
$$- 4 \sin{\left(3 x \right)} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
Consider each factor separately
Step
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$2 x + \frac{\pi}{4} = 2 \pi n + \operatorname{acos}{\left(0 \right)}$$
$$2 x + \frac{\pi}{4} = 2 \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$2 x + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
$$2 x + \frac{\pi}{4} = 2 \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation with the opposite sign, in total:
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n - \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{8}$$
$$x = \pi n - \frac{3 \pi}{8}$$
Step
$$\sin{\left(3 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(3 x \right)} = 0$$
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$3 x = 2 \pi n$$
$$3 x = 2 \pi n + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{8}$$
$$x = \pi n - \frac{3 \pi}{8}$$
$$x = \frac{2 \pi n}{3}$$
$$x = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
Step
$$\sin{\left(x + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x + \frac{\pi}{4} = 2 \pi n$$
$$x + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation with the opposite sign, in total:
$$x = 2 \pi n - \frac{\pi}{4}$$
$$x = 2 \pi n + \frac{3 \pi}{4}$$
The final answer:
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n - \frac{3 \pi}{8}$$
$$x_{3} = \frac{2 \pi n}{3}$$
$$x_{4} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
$$x_{5} = 2 \pi n - \frac{\pi}{4}$$
$$x_{6} = 2 \pi n + \frac{3 \pi}{4}$$
Sum and product of roots
[src]
sum
-7*pi -2*pi -3*pi -pi -pi pi pi 5*pi 2*pi 3*pi
0 + ----- + ----- + ----- + ---- + ---- + -- + -- + ---- + ---- + ---- + pi
8 3 8 3 4 8 3 8 3 4
$$\left(0\right) + \left(- \frac{7 \pi}{8}\right) + \left(- \frac{2 \pi}{3}\right) + \left(- \frac{3 \pi}{8}\right) + \left(- \frac{\pi}{3}\right) + \left(- \frac{\pi}{4}\right) + \left(\frac{\pi}{8}\right) + \left(\frac{\pi}{3}\right) + \left(\frac{5 \pi}{8}\right) + \left(\frac{2 \pi}{3}\right) + \left(\frac{3 \pi}{4}\right) + \left(\pi\right)$$
=
pi
$$\pi$$
product
-7*pi -2*pi -3*pi -pi -pi pi pi 5*pi 2*pi 3*pi
0 * ----- * ----- * ----- * ---- * ---- * -- * -- * ---- * ---- * ---- * pi
8 3 8 3 4 8 3 8 3 4
$$\left(0\right) * \left(- \frac{7 \pi}{8}\right) * \left(- \frac{2 \pi}{3}\right) * \left(- \frac{3 \pi}{8}\right) * \left(- \frac{\pi}{3}\right) * \left(- \frac{\pi}{4}\right) * \left(\frac{\pi}{8}\right) * \left(\frac{\pi}{3}\right) * \left(\frac{5 \pi}{8}\right) * \left(\frac{2 \pi}{3}\right) * \left(\frac{3 \pi}{4}\right) * \left(\pi\right)$$
=
0
$$0$$
Rapid solution
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x_1 = 0
$$x_{1} = 0$$
-7*pi
x_2 = -----
8
$$x_{2} = - \frac{7 \pi}{8}$$
-2*pi
x_3 = -----
3
$$x_{3} = - \frac{2 \pi}{3}$$
-3*pi
x_4 = -----
8
$$x_{4} = - \frac{3 \pi}{8}$$
-pi
x_5 = ----
3
$$x_{5} = - \frac{\pi}{3}$$
-pi
x_6 = ----
4
$$x_{6} = - \frac{\pi}{4}$$
pi
x_7 = --
8
$$x_{7} = \frac{\pi}{8}$$
pi
x_8 = --
3
$$x_{8} = \frac{\pi}{3}$$
5*pi
x_9 = ----
8
$$x_{9} = \frac{5 \pi}{8}$$
2*pi
x_10 = ----
3
$$x_{10} = \frac{2 \pi}{3}$$
3*pi
x_11 = ----
4
$$x_{11} = \frac{3 \pi}{4}$$
x_12 = pi
$$x_{12} = \pi$$
Numerical answer
[src]
x1 = -100.138265833175
x2 = 85.870199198121
x3 = 33.5103216382911
x4 = 63.8790506229925
x5 = 90.0589894029074
x6 = -61.7846555205993
x7 = 80.5033117482384
x8 = -21.9911485751286
x9 = 38.0918109247762
x10 = -17.8023583703422
x11 = 26.1799387799149
x12 = 14.5298660228528
x13 = 96.2112750161874
x14 = -64.009950316892
x15 = -54.1924732744239
x16 = 30.2378292908018
x17 = 28.2743338823081
x18 = -19.8967534727354
x19 = -2.0943951023932
x20 = 43.9822971502571
x21 = -13.7444678594553
x22 = -25.9181393921158
x23 = 8.37758040957278
x24 = -32.2013246992954
x25 = -41.8879020478639
x26 = 2.0943951023932
x27 = -51.4435797025329
x28 = 94.2477796076938
x29 = 21.9911485751286
x30 = 74.2201264410589
x31 = 6.28318530717959
x32 = 4.18879020478639
x33 = 19.8967534727354
x34 = -39.7935069454707
x35 = -3.92699081698724
x36 = 8.24668071567321
x37 = -65.9734457253857
x38 = -7.46128255227576
x39 = 70.162235930172
x40 = 0.0
x41 = -93.8550805259951
x42 = 16.1006623496477
x43 = -81.6814089933346
x44 = -56.1559686829176
x45 = 62.0464549083984
x46 = -79.717913584841
x47 = 40.0553063332699
x48 = -83.7758040957278
x49 = -46.0766922526503
x50 = -59.6902604182061
x51 = 77.4926187885482
x52 = 24.0855436775217
x53 = -49.872783375738
x54 = -34.5575191894877
x55 = 99.4837673636768
x56 = -47.9092879672443
x57 = 82.0741080750334
x58 = 34.9502182711865
x59 = 50.2654824574367
x60 = -95.42587685279
x61 = 11.7809724509617
x62 = 92.1533845053006
x63 = -68.0678408277789
x64 = 74.6128255227576
x65 = 58.5121631731099
x66 = 87.9645943005142
x67 = -76.1836218495525
x68 = -71.8639319508665
x69 = -12.5663706143592
x70 = 46.0766922526503
x71 = -87.9645943005142
x72 = -37.6991118430775
x73 = 60.0829594999048
x74 = -73.4347282776614
x75 = 68.0678408277789
x76 = -35.7356164345839
x77 = 84.037603483527
x78 = -27.8816348006094
x79 = -98.174770424681
x80 = -5.89048622548086
x81 = -43.9822971502571
x82 = 48.1710873550435
x83 = -86.0010988920206
x84 = -57.7267650097125
x85 = 18.0641577581413
x86 = -29.4524311274043
x87 = -10.2101761241668
x88 = -15.707963267949
x89 = 55.5014702134197
x90 = -90.0589894029074
x91 = -91.8915851175014
x92 = 36.5210145979813
x93 = 72.2566310325652
x94 = 41.8879020478639
x95 = 52.2289778659303
x96 = -24.0855436775217
x97 = 65.9734457253857
x98 = -69.9004365423729
x99 = -78.1471172580461
x99 = -78.1471172580461
The graph