Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 3x-2(3x+4)=10
-
Equation (x-1)^3=-8
-
Equation sin(x)=1/5
-
Equation 1*(x-1)*(x-3)=0
- Express {x} in terms of y where:
- 7*x-1*y=0
- -2*x+14*y=9
- 18*x+20*y=-2
- -2*x+7*y=-14
-
Identical expressions
- cos4x- one : two = zero
- co sinus of e of 4x minus 1:2 equally 0
- co sinus of e of 4x minus one : two equally zero
- cos4x-1:2=O
-
Similar expressions
- cos4x+1:2=0
cos4x-1:2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(4 x \right)} - \frac{1}{2} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(4 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$4 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$4 x = \pi n + \frac{\pi}{3}$$
$$4 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$4$$
we get the answer:
$$x_{1} = \frac{\pi n}{4} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{6}$$
$$\cos{\left(4 x \right)} - \frac{1}{2} = 0$$
- this is the simplest trigonometric equation
Move -1/2 to right part of the equation
with the change of sign in -1/2
We get:
$$\cos{\left(4 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$4 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$4 x = \pi n + \frac{\pi}{3}$$
$$4 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$4$$
we get the answer:
$$x_{1} = \frac{\pi n}{4} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{6}$$
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 12 12
$$\frac{\pi}{12} + \frac{5 \pi}{12}$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
pi 5*pi --*---- 12 12
$$\frac{\pi}{12} \frac{5 \pi}{12}$$
=
2 5*pi ----- 144
$$\frac{5 \pi^{2}}{144}$$
5*pi^2/144
Rapid solution
[src]
pi
x1 = --
12
$$x_{1} = \frac{\pi}{12}$$
5*pi
x2 = ----
12
$$x_{2} = \frac{5 \pi}{12}$$
x2 = 5*pi/12
Numerical answer
[src]
x1 = 78.2780169519457
x2 = -78.2780169519457
x3 = 88.2263936883134
x4 = 92.9387826686981
x5 = 48.4328867428426
x6 = -39.5317075576716
x7 = 23.8237442897226
x8 = -30.1069295969022
x9 = 4.45058959258554
x10 = 9.16297857297023
x11 = 74.0892267471593
x12 = -71.9948316447661
x13 = 6.02138591938044
x14 = -37.9609112308767
x15 = 96.0803753222878
x16 = 93.9859802198946
x17 = -81.9432083811338
x18 = 62.5700536839967
x19 = 67.8060414399797
x20 = -64.6644487863899
x21 = -74.0892267471593
x22 = -42.1497014356631
x23 = -100.269165527074
x24 = -83.5140047079287
x25 = -9.68657734856853
x26 = -89.7971900151083
x27 = -96.0803753222878
x28 = -45.2912940892529
x29 = -61.5228561328001
x30 = 87.7027949127151
x31 = -79.8488132787406
x32 = 64.1408500107916
x33 = -21.7293491873294
x34 = -67.8060414399797
x35 = -23.8237442897226
x36 = 1448.53601269269
x37 = 52.0980781720307
x38 = -75.6600230739542
x39 = -15.9697626557481
x40 = 126.972703082587
x41 = 45.8148928648512
x42 = 26.4417381677141
x43 = -34.8193185772869
x44 = 42.1497014356631
x45 = -1.83259571459405
x46 = -52.0980781720307
x47 = 22.2529479629277
x48 = -97.6511716490827
x49 = 100.269165527074
x50 = -87.7027949127151
x51 = 50.0036830696375
x52 = 1.83259571459405
x53 = -65.7116463375865
x54 = -13.8753675533549
x55 = -31.6777259236971
x56 = -59.9520598060052
x57 = -45.8148928648512
x58 = 468.359104772678
x59 = 8.11578102177363
x60 = 85.0848010347236
x61 = -86.1319985859202
x62 = 28.012534494509
x63 = 44.2440965380563
x64 = 0.261799387799149
x65 = -28.012534494509
x66 = -43.720497762458
x67 = 34.2957198016886
x68 = 66.2352451131848
x69 = -57.857664703612
x70 = 56.2868683768171
x71 = 20.1585528605345
x72 = -17.540558982543
x73 = -20.1585528605345
x74 = 71.9948316447661
x75 = -50.0036830696375
x76 = 30.1069295969022
x77 = -6.02138591938044
x78 = 89.7971900151083
x79 = 59.9520598060052
x80 = 86.1319985859202
x81 = 37.9609112308767
x82 = 92.4151838930998
x83 = 81.9432083811338
x84 = 12.30457122656
x85 = -35.8665161284835
x86 = -64.1408500107916
x87 = 15.9697626557481
x88 = 70.4240353179712
x89 = -53.6688744988256
x90 = -93.9859802198946
x91 = -12.8281700021583
x91 = -12.8281700021583