Mister Exam
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How to use it?
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Identical expressions
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Similar expressions
- 2*cos*2*(x)+1=0
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- 2cos^2(x)+1=2sqrt(2)cos((3pi/2)-x)
- √2cos(2*x)+1=0
2*cos(2*x)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \cos{\left(2 x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(2 x \right)} = -1$$
The equation is transformed to
$$\cos{\left(2 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \frac{2 \pi}{3}$$
$$2 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{3}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{6}$$
$$2 \cos{\left(2 x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$2 \cos{\left(2 x \right)} = -1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(2 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \frac{2 \pi}{3}$$
$$2 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{3}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{6}$$
Rapid solution
[src]
pi
x1 = --
3
$$x_{1} = \frac{\pi}{3}$$
2*pi
x2 = ----
3
$$x_{2} = \frac{2 \pi}{3}$$
x2 = 2*pi/3
Sum and product of roots
[src]
sum
pi 2*pi -- + ---- 3 3
$$\frac{\pi}{3} + \frac{2 \pi}{3}$$
=
pi
$$\pi$$
product
pi 2*pi --*---- 3 3
$$\frac{\pi}{3} \frac{2 \pi}{3}$$
=
2 2*pi ----- 9
$$\frac{2 \pi^{2}}{9}$$
2*pi^2/9
Numerical answer
[src]
x1 = -33.5103216382911
x2 = -63.8790506229925
x3 = -79.5870138909414
x4 = -35.6047167406843
x5 = -14.6607657167524
x6 = -4.18879020478639
x7 = -46.0766922526503
x8 = -93.2005820564972
x9 = -5.23598775598299
x10 = -76.4454212373516
x11 = 90.0589894029074
x12 = 98.4365698124802
x13 = -85.870199198121
x14 = 30.3687289847013
x15 = 16.7551608191456
x16 = 55.5014702134197
x17 = -57.5958653158129
x18 = 33.5103216382911
x19 = 61.7846555205993
x20 = 70.162235930172
x21 = 76.4454212373516
x22 = -92.1533845053006
x23 = -70.162235930172
x24 = 39.7935069454707
x25 = 4.18879020478639
x26 = -61.7846555205993
x27 = 24.0855436775217
x28 = -39.7935069454707
x29 = 38.7463093942741
x30 = -17.8023583703422
x31 = 8.37758040957278
x32 = -49.2182849062401
x33 = -99.4837673636768
x34 = 68.0678408277789
x35 = -30.3687289847013
x36 = -60.7374579694027
x37 = -90.0589894029074
x38 = 54.4542726622231
x39 = 60.7374579694027
x40 = 85.870199198121
x41 = -19.8967534727354
x42 = 17.8023583703422
x43 = 99.4837673636768
x44 = 46.0766922526503
x45 = -11.5191730631626
x46 = -294.262511886244
x47 = -77.4926187885482
x48 = 92.1533845053006
x49 = 19.8967534727354
x50 = -13.6135681655558
x51 = -26.1799387799149
x52 = 63.8790506229925
x53 = 74.3510261349584
x54 = 52.3598775598299
x55 = -83.7758040957278
x56 = 77.4926187885482
x57 = 10.471975511966
x58 = -96.342174710087
x59 = -2.0943951023932
x60 = -68.0678408277789
x61 = 79.5870138909414
x62 = 20.943951023932
x63 = -54.4542726622231
x64 = -10.471975511966
x65 = -378.038315981972
x66 = -32.4631240870945
x67 = -41.8879020478639
x68 = -24.0855436775217
x69 = 2.0943951023932
x70 = 1515.29485658148
x71 = 41.8879020478639
x72 = 32.4631240870945
x73 = 96.342174710087
x74 = -71.2094334813686
x75 = -27.2271363311115
x76 = 26.1799387799149
x77 = -98.4365698124802
x78 = 83.7758040957278
x79 = 48.1710873550435
x80 = -55.5014702134197
x81 = 82.7286065445312
x82 = 11.5191730631626
x83 = -48.1710873550435
x83 = -48.1710873550435
The graph