Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 3*x^2-12*x+12=0
-
Equation 5(x-2,2)=7x
-
Equation -2x=4
-
Equation 10/(6-x)=4/(x+2)
- Express {x} in terms of y where:
- 4*x+5*y=7
- 4*x-2*y=4
- -19*x+10*y=11
- 16*x-9*y=-17
-
Identical expressions
- two *sin(3x)+ one = zero
- 2 multiply by sinus of (3x) plus 1 equally 0
- two multiply by sinus of (3x) plus one equally zero
- 2sin(3x)+1=0
- 2sin3x+1=0
- 2*sin(3x)+1=O
-
Similar expressions
- 2*sin(3x)-1=0
2*sin(3x)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \sin{\left(3 x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \sin{\left(3 x \right)} = -1$$
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:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{7 \pi}{18}$$
$$2 \sin{\left(3 x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$2 \sin{\left(3 x \right)} = -1$$
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:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{7 \pi}{18}$$
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
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
Numerical answer
[src]
x1 = 32.637657012294
x2 = -61.6101225953998
x3 = -2.26892802759263
x4 = 61.9591884457987
x5 = -35.7792496658838
x6 = -65.7989128001862
x7 = 94.0732466824944
x8 = 76.6199541625511
x9 = -0.174532925199433
x10 = -86.0447321233204
x11 = -75.5727566113545
x12 = -72.0820981073658
x13 = 30.5432619099008
x14 = 96.1676417848876
x15 = 45.9021593274509
x16 = -11.3446401379631
x17 = -26.0054058547155
x18 = 56.3741348394168
x19 = -54.6288055874225
x20 = -23.9110107523223
x21 = -88.1391272257137
x22 = 50.0909495322373
x23 = -74.176493209759
x24 = -19.7222205475359
x25 = 646.2954220135
x26 = 59.8647933434055
x27 = -48.3456202802429
x28 = -16.929693744345
x29 = -13.4390352403563
x30 = 4.01425727958696
x31 = 28.4488668075076
x32 = 1.91986217719376
x33 = 87.7900613753148
x34 = -21.8166156499291
x35 = 72.4311639577647
x36 = 20.0712863979348
x37 = 6.10865238198015
x38 = 100.356431989674
x39 = 89.884456477708
x40 = -4.36332312998582
x41 = -17.6278254451427
x42 = 41.7133691226645
x43 = 43.8077642250577
x44 = -98.6111027376796
x45 = 26.3544717051144
x46 = -77.6671517137477
x47 = 17.9768912955416
x48 = -79.7615468161409
x49 = 54.2797397370236
x50 = 70.3367688553715
x51 = 64.0535835481919
x52 = 98.2620368872808
x53 = -69.9877030049726
x54 = -33.6848545634906
x55 = -15.5334303427495
x56 = -63.704517697793
x57 = 24.2600766027212
x58 = 68.2423737529783
x59 = -67.8933079025794
x60 = -90.2335223281068
x61 = -51.1381470834339
x62 = 10.2974425867665
x63 = -42.0624349730633
x64 = -44.1568300754565
x65 = -81.8559419185341
x66 = 12.3918376891597
x67 = 83.6012711705284
x68 = 66.1479786505851
x69 = 15.8824961931484
x70 = -59.5157274930066
x71 = -7.15584993317675
x72 = -92.3279174305
x73 = 74.5255590601579
x74 = 85.6956662729216
x75 = -57.4213323906134
x76 = 22.165681500328
x77 = 52.1853446346305
x78 = -31.5904594610974
x79 = -37.873644768277
x80 = 77.3180858633488
x81 = -83.9503370209273
x82 = -39.9680398706701
x83 = -46.2512251778497
x84 = 39.6189740202713
x85 = 91.9788515801012
x86 = -28.0998009571087
x87 = 8.20304748437335
x88 = 47.9965544298441
x89 = 33.3357887130917
x89 = 33.3357887130917