Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation x^4-15x^2-16=0
-
Equation sin(6*x-pi/6)=1/2
- Equation x^2-14/x-10x/x^2-14=3
-
Equation 3x-11=6x-4(2x-1)
- Express {x} in terms of y where:
- 9*x+18*y=6
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- 5*x+5*y=20
- Sum of series:
-
1/2
- Integral of d{x}:
-
1/2
- Derivative of:
-
1/2
-
Identical expressions
- sin(six *x-pi/ six)= one / two
- sinus of (6 multiply by x minus Pi divide by 6) equally 1 divide by 2
- sinus of (six multiply by x minus Pi divide by six) equally one divide by two
- sin(6x-pi/6)=1/2
- sin6x-pi/6=1/2
- sin(6*x-pi divide by 6)=1 divide by 2
-
Similar expressions
- sin(6*x+pi/6)=1/2
-
What you mean?
- sin((6*x - pi)/6) = 1/2
- sin(6*(x - pi)/6) = 1/2
- sin(6*x - pi/6) = 1/2
- sin(6*x - pi/6) = 1/2
You entered:
sin(6*x-pi/6)=1/2
What you mean?
sin(6*x-pi/6)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ pi\ sin|6*x - --| = 1/2 \ 6 /
$$\sin{\left(6 x - \frac{\pi}{6} \right)} = \frac{1}{2}$$
Detail solution
Given the equation
$$\sin{\left(6 x - \frac{\pi}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $-1$
The equation is transformed to
$$\cos{\left(6 x + \frac{\pi}{3} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$6 x + \frac{\pi}{3} = 2 \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$6 x + \frac{\pi}{3} = 2 \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$6 x + \frac{\pi}{3} = 2 \pi n + \frac{2 \pi}{3}$$
$$6 x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation with the opposite sign, in total:
$$6 x = 2 \pi n + \frac{\pi}{3}$$
$$6 x = 2 \pi n - \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$6$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{18}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{9}$$
$$\sin{\left(6 x - \frac{\pi}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $-1$
The equation is transformed to
$$\cos{\left(6 x + \frac{\pi}{3} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$6 x + \frac{\pi}{3} = 2 \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$6 x + \frac{\pi}{3} = 2 \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$6 x + \frac{\pi}{3} = 2 \pi n + \frac{2 \pi}{3}$$
$$6 x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation with the opposite sign, in total:
$$6 x = 2 \pi n + \frac{\pi}{3}$$
$$6 x = 2 \pi n - \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$6$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{18}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{9}$$
Rapid solution
[src]
pi
x_1 = --
18
$$x_{1} = \frac{\pi}{18}$$
pi
x_2 = --
6
$$x_{2} = \frac{\pi}{6}$$
Sum and product of roots
[src]
sum
pi pi -- + -- 18 6
$$\left(\frac{\pi}{18}\right) + \left(\frac{\pi}{6}\right)$$
=
2*pi ---- 9
$$\frac{2 \pi}{9}$$
product
pi pi -- * -- 18 6
$$\left(\frac{\pi}{18}\right) * \left(\frac{\pi}{6}\right)$$
=
2 pi --- 108
$$\frac{\pi^{2}}{108}$$
Numerical answer
[src]
x1 = 39.9680398706701
x2 = -91.9788515801012
x3 = -71.733032256967
x4 = 68.2423737529783
x5 = 38.2227106186758
x6 = 4.36332312998582
x7 = 92.3279174305
x8 = -26.0054058547155
x9 = 56.025068989018
x10 = -23.9110107523223
x11 = 53.9306738866248
x12 = 29.4960643587042
x13 = -19.7222205475359
x14 = -39.6189740202713
x15 = -4.01425727958696
x16 = -85.6956662729216
x17 = 60.2138591938044
x18 = 2.26892802759263
x19 = -5.75958653158129
x20 = -43.8077642250577
x21 = -29.845130209103
x22 = 80.1106126665397
x23 = -53.9306738866248
x24 = -73.8274273593601
x25 = 46.2512251778497
x26 = 17.9768912955416
x27 = 94.7713783832921
x28 = -31.9395253114962
x29 = 51.4872129338327
x30 = -7.85398163397448
x31 = 47.2984227290463
x32 = 86.0447321233204
x33 = 21.4675497995303
x34 = -100.007366139275
x35 = -45.9021593274509
x36 = 48.3456202802429
x37 = 73.4783615089613
x38 = 12.0427718387609
x39 = -14.1371669411541
x40 = 16.2315620435473
x41 = -63.704517697793
x42 = -41.7133691226645
x43 = 24.2600766027212
x44 = -61.6101225953998
x45 = -49.7418836818384
x46 = 14.1371669411541
x47 = 44.1568300754565
x48 = 0.174532925199433
x49 = 100.007366139275
x50 = 7.85398163397448
x51 = 66.1479786505851
x52 = -36.1283155162826
x53 = 75.9218224617533
x54 = 20.0712863979348
x55 = 78.0162175641465
x56 = -95.8185759344887
x57 = -87.7900613753148
x58 = -93.7241808320955
x59 = 22.165681500328
x60 = -34.0339204138894
x61 = 97.9129710368819
x62 = 36.1283155162826
x63 = -51.8362787842316
x64 = -78.0162175641465
x65 = -12.0427718387609
x66 = -80.1106126665397
x67 = -75.9218224617533
x68 = 34.0339204138894
x69 = -97.9129710368819
x70 = -21.8166156499291
x71 = -83.6012711705284
x72 = -56.025068989018
x73 = 90.2335223281068
x74 = 88.1391272257137
x75 = -65.7989128001862
x76 = -67.8933079025794
x77 = -47.9965544298441
x78 = -58.1194640914112
x79 = -27.7507351067098
x80 = 70.3367688553715
x81 = 42.0624349730633
x82 = -1.91986217719376
x83 = -89.884456477708
x84 = -9.94837673636768
x85 = 64.0535835481919
x86 = 61.261056745001
x87 = 82.2050077689329
x88 = -17.6278254451427
x89 = -69.9877030049726
x90 = 58.1194640914112
x91 = 31.9395253114962
x92 = 26.3544717051144
x93 = 9.94837673636768
x93 = 9.94837673636768
The graph