Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 2^x=6-x
-
Equation x^3=4
-
Equation x^2+2x+2=0
-
Equation 3/(x-5)+8/x=2
- Express {x} in terms of y where:
- -1*x-6*y=18
- 6*x-19*y=-18
- -14*x+18*y=0
- -8*x+3*y=-4
-
Identical expressions
- two sin2x- one /2= one
- 2 sinus of 2x minus 1 divide by 2 equally 1
- two sinus of 2x minus one divide by 2 equally one
- 2sin2x-1 divide by 2=1
-
Similar expressions
- 2sin2x+1/2=1
2sin2x-1/2=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \sin{\left(2 x \right)} - \frac{1}{2} = 1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{3}{4}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{4} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{4} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{4} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{4} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$
$$x_{2} = \pi n - \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}$$
$$2 \sin{\left(2 x \right)} - \frac{1}{2} = 1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{3}{4}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{4} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{4} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{4} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{4} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$
$$x_{2} = \pi n - \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}$$
Sum and product of roots
[src]
sum
pi asin(3/4) asin(3/4) -- - --------- + --------- 2 2 2
$$\frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \left(- \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}\right)$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
/pi asin(3/4)\ asin(3/4) |-- - ---------|*--------- \2 2 / 2
$$\left(- \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}\right) \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$
=
(pi - asin(3/4))*asin(3/4)
--------------------------
4
$$\frac{\left(\pi - \operatorname{asin}{\left(\frac{3}{4} \right)}\right) \operatorname{asin}{\left(\frac{3}{4} \right)}}{4}$$
(pi - asin(3/4))*asin(3/4)/4
Rapid solution
[src]
pi asin(3/4)
x1 = -- - ---------
2 2
$$x_{1} = - \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2} + \frac{\pi}{2}$$
asin(3/4)
x2 = ---------
2
$$x_{2} = \frac{\operatorname{asin}{\left(\frac{3}{4} \right)}}{2}$$
x2 = asin(3/4)/2
Numerical answer
[src]
x1 = -30.2691612485938
x2 = 48.2706550911511
x3 = -1.99482736628564
x4 = 12.9904016538499
x5 = 0.424031039490741
x6 = -49.841451417946
x7 = 107.960915509357
x8 = -12.1423395748684
x9 = 78.9638473792356
x10 = -59.2662293787153
x11 = -8.27801267346522
x12 = -58.5434951309019
x13 = -14.5611979806448
x14 = 9.84880900026012
x15 = -21.5671175356378
x16 = 100.954995954364
x17 = 72.680662072056
x18 = -52.2603098237223
x19 = 6.70721634667033
x20 = -36.5523465557734
x21 = -83.6762363596203
x22 = 66.3974767648764
x23 = 31.8399575753887
x24 = -17.7027906342346
x25 = 44.4063281897478
x26 = 35.7042844767919
x27 = 60.1142914576968
x28 = -34.133488149997
x29 = -67.9682730916713
x30 = 13.7131359016633
x31 = -43.5582661107664
x32 = 89.1113595878184
x33 = 63.9786183591
x34 = 92.2529522414082
x35 = 19.9963212088429
x36 = 94.6718106471845
x37 = -5.85915426768885
x38 = 82.1054400328254
x39 = -56.1246367251255
x40 = 50.6895134969274
x41 = -2.71756161409905
x42 = -52.9830440715357
x43 = 41.9874697839715
x44 = -78.1157853002541
x45 = 75.8222547256458
x46 = 34.9815502289785
x47 = -100.106933875383
x48 = 4.28835794089395
x49 = -27.8503028428174
x50 = -39.6939392093632
x51 = -30.9918954964072
x52 = -74.2514583988509
x53 = 97.8134033007743
x54 = 85.9697669342286
x55 = -87.5405632610235
x56 = 79.686581627049
x57 = -65.5494146858949
x58 = -23.9859759414142
x59 = 95.3945448949979
x60 = 38.1231428825683
x61 = 22.4151796146193
x62 = 7.42995059448374
x63 = -81.2573779538439
x64 = 26.2795065160225
x65 = -37.2750808035868
x66 = -89.9594216667998
x67 = -93.8237485682031
x68 = 53.8311061505172
x69 = 28.6983649217989
x70 = -80.5346437060305
x71 = -15.2839322284582
x72 = 16.1319943074397
x73 = 56.972698804107
x74 = -61.6850877844917
x75 = -71.8325999930745
x76 = 51.4122477447408
x77 = -96.2426069739794
x78 = -45.9771245165427
x79 = 29.4210991696123
x80 = 73.4033963198694
x81 = 57.6954330519204
x82 = -84.3989706074337
x83 = 70.2618036662796
x84 = 88.3886253400049
x85 = -9.00074692127864
x85 = -9.00074692127864