Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x+10)^2=(5-x)^2
-
Equation x^2+x-20=0
-
Equation x^5-x^4-2x^3+2x^2-3x+3=0
- Equation 4y^3-(4y^3-8y)-(6y+3)=7
- Express {x} in terms of y where:
- -10*x+18*y=6
- 9*x-18*y=6
- -1*x+19*y=-10
- 4*x+13*y=13
- Derivative of:
- -0,5
-
Identical expressions
- sin(3x- one)- one =- zero , five
- sinus of (3x minus 1) minus 1 equally minus 0,5
- sinus of (3x minus one) minus one equally minus zero , five
- sin3x-1-1=-0,5
-
Similar expressions
- 144-(0,5625*0,5625+1,125x+1.5625x*x)=x*x+2.5x+1.5625
- sin(3x+1)-1=-0,5
- sin(3x-1)-1=+0,5
- sin(3x-1)+1=-0,5
sin(3x-1)-1=-0,5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(3 x - 1 \right)} - 1 = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x - 1 = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$3 x - 1 = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$3 x - 1 = 2 \pi n + \frac{\pi}{6}$$
$$3 x - 1 = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$-1$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n + \frac{\pi}{6} + 1$$
$$3 x = 2 \pi n + 1 + \frac{5 \pi}{6}$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{18} + \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{1}{3} + \frac{5 \pi}{18}$$
$$\sin{\left(3 x - 1 \right)} - 1 = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x - 1 = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$3 x - 1 = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$3 x - 1 = 2 \pi n + \frac{\pi}{6}$$
$$3 x - 1 = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$-1$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n + \frac{\pi}{6} + 1$$
$$3 x = 2 \pi n + 1 + \frac{5 \pi}{6}$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{18} + \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{1}{3} + \frac{5 \pi}{18}$$
Sum and product of roots
[src]
sum
1 5*pi 1 pi - + ---- + - + -- 3 18 3 18
$$\left(\frac{\pi}{18} + \frac{1}{3}\right) + \left(\frac{1}{3} + \frac{5 \pi}{18}\right)$$
=
2 pi - + -- 3 3
$$\frac{2}{3} + \frac{\pi}{3}$$
product
/1 5*pi\ /1 pi\ |- + ----|*|- + --| \3 18 / \3 18/
$$\left(\frac{1}{3} + \frac{5 \pi}{18}\right) \left(\frac{\pi}{18} + \frac{1}{3}\right)$$
=
(6 + pi)*(6 + 5*pi)
-------------------
324
$$\frac{\left(6 + 5 \pi\right) \left(\pi + 6\right)}{324}$$
(6 + pi)*(6 + 5*pi)/324
Rapid solution
[src]
1 5*pi
x1 = - + ----
3 18
$$x_{1} = \frac{1}{3} + \frac{5 \pi}{18}$$
1 pi
x2 = - + --
3 18
$$x_{2} = \frac{\pi}{18} + \frac{1}{3}$$
x2 = pi/18 + 1/3
Numerical answer
[src]
x1 = 80.0948801494742
x2 = 46.5845585111831
x3 = 22.1499489832625
x4 = -21.8323481669947
x5 = -72.0978306244313
x6 = 92.6612507638334
x7 = -53.9464064036903
x8 = 0.507866258532766
x9 = -61.6258551124654
x10 = -5.77531904864682
x11 = -1.58652884386043
x12 = 1548.2658469271
x13 = 75.9060899446878
x14 = 86.3780654566538
x15 = -56.0408015060835
x16 = 5.39478816411689
x17 = -40.6819040885334
x18 = 17.9611587784761
x19 = 96.8500409686198
x20 = 40.3013732040035
x21 = 31.9237927944307
x22 = 70.3210363383059
x23 = -34.3987187813538
x24 = -3.68092394625362
x25 = 24.2443440856556
x26 = -12.0585043558264
x27 = 2.60226136092596
x28 = 52.8677438183627
x29 = 13.7723685736897
x30 = 28.433134290442
x31 = -74.1922257268245
x32 = 68.2266412359128
x33 = -76.2866208292177
x34 = -45.5688259941175
x35 = -100.023098656341
x36 = -30.2099285765674
x37 = 99.6425677718107
x38 = -89.5511231443746
x39 = -84.6642012387905
x40 = 50.7733487159695
x41 = 72.4154314406992
x42 = 66.1322461335196
x43 = -58.1351966084767
x44 = 64.0378510311264
x45 = 26.3387391880488
x46 = 78.000485047081
x47 = 13.0742368728919
x48 = -49.7576161989039
x49 = -441.409500346431
x50 = -65.8146453172518
x51 = -91.6455182467678
x52 = -67.909040419645
x53 = -78.3810159316109
x54 = -19.7379530646015
x55 = -85.3623329395882
x56 = -9.96410925343321
x57 = 38.2069781016103
x58 = 10.9798417704987
x59 = -70.0034355220381
x60 = -32.3043236789606
x61 = -93.739913349161
x62 = 57.7546657239468
x63 = -28.1155334741742
x64 = 82.1892752518674
x65 = 11.6779734712965
x66 = -125.155839885059
x67 = 20.0555538808693
x68 = -43.4744308917243
x69 = -17.6435579622083
x70 = 29.8293976920375
x71 = -23.9267432693878
x72 = -14.1528994582196
x73 = 49.377085314374
x74 = -95.8343084515542
x75 = 73.8116948422946
x76 = -87.4567280419814
x77 = 44.4901634087899
x78 = 36.1125829992171
x79 = 59.84906082634
x80 = 84.2836703542606
x81 = -47.6632210965107
x82 = -41.3800357893311
x83 = 310.478341412726
x84 = -63.7202502148586
x85 = 55.6602706215536
x86 = -51.8520113012971
x87 = 42.3957683063967
x88 = 15.8667636760829
x89 = -7.86971415104002
x90 = -26.021138371781
x91 = 88.472460559047
x92 = -97.9287035539474
x93 = 61.9434559287332
x94 = 90.5668556614402
x95 = 34.0181878968239
x95 = 34.0181878968239