Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x^2-8=0
-
Equation x^2-3x+5=0
-
Equation x^2+10*x+24=0
-
Equation (x-8)^2=(x-2)^2
- Express {x} in terms of y where:
- -14*x-3*y=-14
- 4*x-8*y=10
- 18*x-7*y=3
- -13*x+16*y=18
- Sum of series:
-
-1
- Derivative of:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- 2sin(five *x)=- one
- 2 sinus of (5 multiply by x) equally minus 1
- 2 sinus of (five multiply by x) equally minus one
- 2sin(5x)=-1
- 2sin5x=-1
-
Similar expressions
- 2*sin(5*x)=-4
- 2sin(5*x)=+1
- 1/2*(sin(5*x-x)+sin(5*x+x))*1/2*(sin(x-5*x)+sin(x+5*x))=0
- 2*(sin(5x))*(sin((3x)/2))=sin(x/2)
- (√2+2cos7x)(2sin5x+1)=0
2sin(5*x)=-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \sin{\left(5 x \right)} = -1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(5 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$5 x = 2 \pi n - \frac{\pi}{6}$$
$$5 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{5} - \frac{\pi}{30}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{7 \pi}{30}$$
$$2 \sin{\left(5 x \right)} = -1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(5 x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$5 x = 2 \pi n - \frac{\pi}{6}$$
$$5 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{5} - \frac{\pi}{30}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{7 \pi}{30}$$
Rapid solution
[src]
-pi
x1 = ----
30
$$x_{1} = - \frac{\pi}{30}$$
7*pi
x2 = ----
30
$$x_{2} = \frac{7 \pi}{30}$$
x2 = 7*pi/30
Sum and product of roots
[src]
sum
pi 7*pi - -- + ---- 30 30
$$- \frac{\pi}{30} + \frac{7 \pi}{30}$$
=
pi -- 5
$$\frac{\pi}{5}$$
product
-pi 7*pi ----*---- 30 30
$$- \frac{\pi}{30} \frac{7 \pi}{30}$$
=
2 -7*pi ------ 900
$$- \frac{7 \pi^{2}}{900}$$
-7*pi^2/900
Numerical answer
[src]
x1 = 22.0958683302482
x2 = 26.2846585350346
x3 = 6.17846555205993
x4 = 8.27286065445312
x5 = -69.6386371545737
x6 = -91.8392252399416
x7 = 30.0545697193424
x8 = 92.8864227911382
x9 = 32.1489648217356
x10 = -8.0634211442138
x11 = 96.2374549549673
x12 = -44.0870169053768
x13 = -81.7861287484543
x14 = -88.0693140556339
x15 = -0.10471975511966
x16 = -99.7979266290358
x17 = -31.5206462910176
x18 = -49.1135651511204
x19 = 86.18435846348
x20 = 66.0781654805053
x21 = -41.9926218029836
x22 = 94.1430598525741
x23 = -89.7448301375484
x24 = -5.55014702134197
x25 = 9.52949771588904
x26 = -34.0339204138894
x27 = -25.6563400043166
x28 = -10.157816246607
x29 = -35.709436495804
x30 = 87.8598745453945
x31 = -62.9365728269155
x32 = -91.0014671989843
x33 = 72.3613507876849
x34 = -59.5855406630864
x35 = -52.0457182944709
x36 = 89.9542696477877
x37 = -78.0162175641465
x38 = 1.98967534727354
x39 = 45.9719724975306
x40 = -67.9631210726592
x41 = -71.733032256967
x42 = 53.9306738866248
x43 = 82.4144472791722
x44 = -23.9808239224021
x45 = 60.2138591938044
x46 = -55.8156294787787
x47 = 74.0368668695995
x48 = 40.1076662108297
x49 = -57.9100245811719
x50 = -45.7625329872913
x51 = 20.001473227855
x52 = 63.9837703781121
x53 = -31.9395253114962
x54 = -15.6032435128293
x55 = -98.1224105471212
x56 = 12.0427718387609
x57 = -13.9277274309148
x58 = -85.9749189532407
x59 = 77.8067780539072
x60 = 50.160762702317
x61 = -72.1519112774456
x62 = -17.6976386152225
x63 = -20.629791758573
x64 = -75.9218224617533
x65 = -11.8333323285216
x66 = -96.028015444728
x67 = 23.3525053916841
x68 = 76.1312619719927
x69 = -65.868725970266
x70 = -1.78023583703422
x71 = 28.3790536374278
x72 = 43.8775773951374
x73 = 9.94837673636768
x74 = -47.8569280896845
x75 = 56.025068989018
x76 = 100.007366139275
x77 = 97.9129710368819
x78 = 38.4321501289151
x79 = -21.8864288200089
x80 = -54.1401133968641
x81 = -3.87463093942741
x82 = -27.7507351067098
x83 = 62.3082542961976
x84 = 52.2551578047102
x85 = 42.2020613132229
x86 = -37.8038315981972
x87 = 50.9985207432743
x88 = -79.6917336460611
x89 = 16.2315620435473
x90 = 84.0899633610868
x91 = -64.1932098883514
x92 = 33.8244809036501
x93 = 18.3259571459405
x94 = 70.2669556852917
x94 = 70.2669556852917