Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation cos(x)=3/4
-
Equation x^2-7x=0
-
Equation x^2+21=10x
-
Equation sin5x=0
- Express {x} in terms of y where:
- 16*x+7*y=8
- 18*x-20*y=-7
- 15*x-3*y=19
- -11*x+6*y=4
- Derivative of:
-
sin(6x)
- Integral of d{x}:
- sin(6x)
- Sum of series:
-
(-1/2)
-
Identical expressions
- sin(6x)=(- one / two)
- sinus of (6x) equally ( minus 1 divide by 2)
- sinus of (6x) equally ( minus one divide by two)
- sin6x=-1/2
- sin(6x)=(-1 divide by 2)
-
Similar expressions
- sin6x+sin2x+2sin^2x=1
- 3*sin(x)^(3)-(5/2)*sin(6*x)+1=0
- sin(5x)=-1/2
- 3*9^(x-1/2)-7*6^x+3*4^(x+1)=0
- sin(6x)=(1/2)
- sin(x)-1/2=0
- sint=-1/2
- cos4x*cos2x=sin6x*sin8x
sin(6x)=(-1/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(6 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$6 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$6 x = 2 \pi n - \frac{\pi}{6}$$
$$6 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} + \frac{7 \pi}{36}$$
$$\sin{\left(6 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$6 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$6 x = 2 \pi n - \frac{\pi}{6}$$
$$6 x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} + \frac{7 \pi}{36}$$
Rapid solution
[src]
-pi
x_1 = ----
36
$$x_{1} = - \frac{\pi}{36}$$
7*pi
x_2 = ----
36
$$x_{2} = \frac{7 \pi}{36}$$
Sum and product of roots
[src]
sum
-pi 7*pi ---- + ---- 36 36
$$\left(- \frac{\pi}{36}\right) + \left(\frac{7 \pi}{36}\right)$$
=
pi -- 6
$$\frac{\pi}{6}$$
product
-pi 7*pi ---- * ---- 36 36
$$\left(- \frac{\pi}{36}\right) * \left(\frac{7 \pi}{36}\right)$$
=
2 -7*pi ------ 1296
$$- \frac{7 \pi^{2}}{1296}$$
Numerical answer
[src]
x1 = -22.0784150377283
x2 = -37.7863783056772
x3 = -61.871921983199
x4 = 23.998277214922
x5 = 43.8950306876574
x6 = -85.9574656607207
x7 = -97.8257045742822
x8 = 45.9894257900506
x9 = -88.0518607631139
x10 = -71.2966999439684
x11 = -63.9663170855922
x12 = 70.0749694675723
x13 = -3.22885911618951
x14 = -66.0607121879854
x15 = 96.2549082474873
x16 = -46.5130245656489
x17 = 2.00712863979348
x18 = 100.094632601875
x19 = -80.02334620394
x20 = -27.6634686441101
x21 = 40.7534380340676
x22 = -7.76671517137477
x23 = 62.3955207587973
x24 = -11.9555053761612
x25 = 14.2244334037538
x26 = -29.7578637465033
x27 = 92.0661180427009
x28 = 82.2922742315326
x29 = -15.7952297305487
x30 = -31.8522588488965
x31 = 80.1978791291394
x32 = 12.1300383013606
x33 = 4.10152374218667
x34 = -81.7686754559343
x35 = 84.3866693339258
x36 = -33.9466539512897
x37 = -53.8434074240251
x38 = -68.5041731407774
x39 = 18.762289458939
x40 = 52.2726110972302
x41 = -75.8345559991536
x42 = -99.9200996766754
x43 = -77.9289511015468
x44 = -41.9751685104636
x45 = -71.6457657943672
x46 = -24.5218759905203
x47 = 72.1693645699655
x48 = -17.8896248329419
x49 = -36.7391807544806
x50 = 56.1123354516177
x51 = -51.7490123216319
x52 = 6.19591884457987
x53 = 89.9717229403077
x54 = -83.8630705583275
x55 = 26.0926723173152
x56 = 50.178215994837
x57 = 10.0356431989674
x58 = 16.318828506147
x59 = 21.9038821125288
x60 = 499.076899607779
x61 = -93.6369143694958
x62 = 1816.4514190131
x63 = -58.0321976288115
x64 = 30.2814625221016
x65 = 54.0179403492245
x66 = 74.2637596723587
x67 = -5.67232006898157
x68 = 65.8861792627859
x69 = -59.7775268808058
x70 = 36.2155819788823
x71 = -9.86111027376796
x72 = 94.1605131450941
x73 = 60.3011256564041
x74 = -19.9840199353351
x75 = -95.731309471889
x76 = 98.0002374994816
x77 = -0.0872664625997165
x78 = 67.9805743651791
x79 = -90.1462558655071
x80 = 76.0090889243531
x81 = -44.0695636128568
x82 = 38.3099770812755
x83 = 78.1034840267462
x84 = 48.0838208924438
x85 = 32.0267917740959
x86 = -39.8807734080704
x87 = -4.27605666738611
x88 = -55.9378025264183
x89 = -73.7401608967604
x90 = 28.1870674197084
x91 = -49.6546172192387
x92 = 58.2067305540109
x93 = 34.1211868764891
x94 = 87.8773278379145
x94 = 87.8773278379145
The graph