Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^3=27
-
Equation x^4+2*x^2-8=0
-
Equation cos(x)=2/3
-
Equation (x-11)^4=(x+3)^4
- Express {x} in terms of y where:
- 6*x+17*y=3
- 1*x+4*y=20
- 20*x+7*y=-9
- -17*x+18*y=10
- Derivative of:
-
2sin3x
-
-1
- Graphing y =:
- 2sin3x
- Sum of series:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- 2sin3x=- one
- 2 sinus of 3x equally minus 1
- 2 sinus of 3x equally minus one
-
Similar expressions
- 2*sin(3*x-pi/3)=sqrt(3)
- 2sin3x=+1
- sin(4x)+sin(2*x)+2sin(3*x)=0
- 2*sin^3(x)+sin(x)+√2=√2*cos(2x)
- 2*sin(3*x-pi/4)+sqrt(3)=0
2sin3x=-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \sin{\left(3 x \right)} = -1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(3 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$3 x = 2 \pi n - \frac{\pi}{6}$$
$$3 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{7 \pi}{18}$$
$$2 \sin{\left(3 x \right)} = -1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(3 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$3 x = 2 \pi n - \frac{\pi}{6}$$
$$3 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{7 \pi}{18}$$
Sum and product of roots
[src]
sum
pi 7*pi - -- + ---- 18 18
$$- \frac{\pi}{18} + \frac{7 \pi}{18}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
-pi 7*pi ----*---- 18 18
$$- \frac{\pi}{18} \frac{7 \pi}{18}$$
=
2 -7*pi ------ 324
$$- \frac{7 \pi^{2}}{324}$$
-7*pi^2/324
Rapid solution
[src]
-pi
x1 = ----
18
$$x_{1} = - \frac{\pi}{18}$$
7*pi
x2 = ----
18
$$x_{2} = \frac{7 \pi}{18}$$
x2 = 7*pi/18
Numerical answer
[src]
x1 = 1.91986217719376
x2 = 87.7900613753148
x3 = -51.1381470834339
x4 = 10.2974425867665
x5 = -19.7222205475359
x6 = 33.3357887130917
x7 = 91.9788515801012
x8 = -74.176493209759
x9 = 20.0712863979348
x10 = 24.2600766027212
x11 = 22.165681500328
x12 = 54.2797397370236
x13 = 26.3544717051144
x14 = 66.1479786505851
x15 = -92.3279174305
x16 = -59.5157274930066
x17 = 17.9768912955416
x18 = -44.1568300754565
x19 = -23.9110107523223
x20 = 85.6956662729216
x21 = 12.3918376891597
x22 = 41.7133691226645
x23 = 64.0535835481919
x24 = -13.4390352403563
x25 = 77.3180858633488
x26 = -26.0054058547155
x27 = -0.174532925199433
x28 = 70.3367688553715
x29 = 74.5255590601579
x30 = 30.5432619099008
x31 = -79.7615468161409
x32 = -31.5904594610974
x33 = -15.5334303427495
x34 = 72.4311639577647
x35 = -67.8933079025794
x36 = 96.1676417848876
x37 = -37.873644768277
x38 = -21.8166156499291
x39 = -42.0624349730633
x40 = 98.2620368872808
x41 = -90.2335223281068
x42 = 15.8824961931484
x43 = 52.1853446346305
x44 = -54.6288055874225
x45 = -48.3456202802429
x46 = 8.20304748437335
x47 = 45.9021593274509
x48 = -81.8559419185341
x49 = -11.3446401379631
x50 = 59.8647933434055
x51 = -39.9680398706701
x52 = -35.7792496658838
x53 = 94.0732466824944
x54 = 83.6012711705284
x55 = 76.6199541625511
x56 = 43.8077642250577
x57 = -28.0998009571087
x58 = -98.6111027376796
x59 = -4.36332312998582
x60 = -61.6101225953998
x61 = 89.884456477708
x62 = -57.4213323906134
x63 = 50.0909495322373
x64 = -63.704517697793
x65 = -77.6671517137477
x66 = -7.15584993317675
x67 = -75.5727566113545
x68 = 4.01425727958696
x69 = -16.929693744345
x70 = -69.9877030049726
x71 = 6.10865238198015
x72 = 100.356431989674
x73 = 28.4488668075076
x74 = -86.0447321233204
x75 = 68.2423737529783
x76 = 646.2954220135
x77 = -72.0820981073658
x78 = 39.6189740202713
x79 = -33.6848545634906
x80 = 32.637657012294
x81 = -83.9503370209273
x82 = -2.26892802759263
x83 = -46.2512251778497
x84 = -88.1391272257137
x85 = -17.6278254451427
x86 = 61.9591884457987
x87 = 56.3741348394168
x88 = 47.9965544298441
x89 = -65.7989128001862
x89 = -65.7989128001862
The graph