Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2+5x+6=0
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Equation 5x+12=8x+30
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Equation 81*x^3+18*x^2+x=0
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Equation x(x^2+2x+1)=6(x+1)
- Express {x} in terms of y where:
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Identical expressions
- two *sin(two *x)- one = zero
- 2 multiply by sinus of (2 multiply by x) minus 1 equally 0
- two multiply by sinus of (two multiply by x) minus one equally zero
- 2sin(2x)-1=0
- 2sin2x-1=0
- 2*sin(2*x)-1=O
-
Similar expressions
- (10sin^2x+cosx)^3=(2sin^2x-1)^3
- 2sin^2x-11sinx+9=0
- 2*sin(2*x)+1=0
- 2sin2x-1/2=1
2*sin(2*x)-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \sin{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \sin{\left(2 x \right)} = 1$$
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
$$2 \sin{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move -1 to right part of the equation
with the change of sign in -1
We get:
$$2 \sin{\left(2 x \right)} = 1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
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
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
Numerical answer
[src]
x1 = -56.2868683768171
x2 = 86.1319985859202
x3 = 28.5361332701073
x4 = -43.720497762458
x5 = -58.3812634792103
x6 = -1.83259571459405
x7 = 37.9609112308767
x8 = 88.2263936883134
x9 = 45.2912940892529
x10 = 94.5095789954929
x11 = -64.6644487863899
x12 = 29.5833308213039
x13 = -9.16297857297023
x14 = -50.0036830696375
x15 = 66.2352451131848
x16 = 64.1408500107916
x17 = -694.030177055545
x18 = 79.8488132787406
x19 = 20.1585528605345
x20 = 13.8753675533549
x21 = 6.54498469497874
x22 = -96.0803753222878
x23 = -100.269165527074
x24 = 72.5184304203644
x25 = 75.6600230739542
x26 = 4.45058959258554
x27 = -36.3901149040818
x28 = 7.59218224617533
x29 = -6.02138591938044
x30 = 31.6777259236971
x31 = -23.8237442897226
x32 = -75.1364242983559
x33 = -8.11578102177363
x34 = -71.9948316447661
x35 = 35.8665161284835
x36 = 0.261799387799149
x37 = 81.9432083811338
x38 = 42.1497014356631
x39 = 95.5567765466895
x40 = -67.8060414399797
x41 = -65.7116463375865
x42 = -97.1275728734844
x43 = 22.2529479629277
x44 = -34.2957198016886
x45 = -83.5140047079287
x46 = -87.7027949127151
x47 = 26.4417381677141
x48 = -74.0892267471593
x49 = -21.7293491873294
x50 = 70.4240353179712
x51 = 78.801615727544
x52 = 57.857664703612
x53 = -15.4461638801498
x54 = 53.6688744988256
x55 = 51.5744793964324
x56 = -86.6555973615185
x57 = -78.2780169519457
x58 = 15.9697626557481
x59 = 48.4328867428426
x60 = -37.4373124552784
x61 = 92.4151838930998
x62 = -45.8148928648512
x63 = 34.8193185772869
x64 = -80.3724120543389
x65 = 59.9520598060052
x66 = 50.5272818452358
x67 = -52.0980781720307
x68 = 9.68657734856853
x69 = -40.5789051088682
x70 = 73.565627971561
x71 = 12.8281700021583
x72 = -89.7971900151083
x73 = -14.3989663289532
x74 = 44.2440965380563
x75 = -53.1452757232273
x76 = -81.4196096055355
x77 = -28.012534494509
x78 = -61.5228561328001
x79 = -12.30457122656
x80 = -17.540558982543
x81 = 100.792764302673
x82 = -59.4284610304069
x83 = -93.9859802198946
x84 = -55.2396708256205
x85 = -39.5317075576716
x86 = -31.1541271480988
x87 = -30.1069295969022
x88 = 56.8104671524154
x89 = -42.6733002112614
x89 = -42.6733002112614
The graph