Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (5x+2)(-x-4)=0
-
Equation sin(x)=3/4
-
Equation 9-2*(-4*x+7)=7
-
Equation x^2-4x+5=0
- Express {x} in terms of y where:
- -11*x+10*y=-9
- -9*x-4*y=-20
- 4*x-7*y=4
- -6*x-12*y=9
-
Identical expressions
- sin(x/ two -П/ six)+ one = zero
- sinus of (x divide by 2 minus П divide by 6) plus 1 equally 0
- sinus of (x divide by two minus П divide by six) plus one equally zero
- sinx/2-П/6+1=0
- sin(x/2-П/6)+1=O
- sin(x divide by 2-П divide by 6)+1=0
-
Similar expressions
- sin(x/2-П/6)-1=0
- sin(x/2+П/6)+1=0
sin(x/2-П/6)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ sin|- - --| + 1 = 0 \2 6 /
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} + 1 = 0$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = -1$$
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 1$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{4 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n - \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{8 \pi}{3}$$
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = -1$$
Divide both parts of the equation by -1
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 1$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{4 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n - \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{8 \pi}{3}$$
Rapid solution
[src]
-2*pi
x1 = -----
3
$$x_{1} = - \frac{2 \pi}{3}$$
10*pi
x2 = -----
3
$$x_{2} = \frac{10 \pi}{3}$$
x2 = 10*pi/3
Sum and product of roots
[src]
sum
2*pi 10*pi - ---- + ----- 3 3
$$- \frac{2 \pi}{3} + \frac{10 \pi}{3}$$
=
8*pi ---- 3
$$\frac{8 \pi}{3}$$
product
-2*pi 10*pi -----*----- 3 3
$$- \frac{2 \pi}{3} \frac{10 \pi}{3}$$
=
2 -20*pi ------- 9
$$- \frac{20 \pi^{2}}{9}$$
-20*pi^2/9
Numerical answer
[src]
x1 = -2.09439439257737
x2 = 10.4719762206937
x3 = 23.0383451201825
x4 = -14.6607653578844
x5 = -27.2271353019155
x6 = -27.227135734615
x7 = 35.6047170574458
x8 = -90.0589903445966
x9 = 10.4719749352438
x10 = 23.0383457361778
x11 = 23.0383441118675
x12 = -77.4926196986972
x13 = -27.2271363694929
x14 = 10.4719765014178
x15 = -39.7935060417991
x16 = 73.3038275589692
x17 = -64.9262491841222
x18 = 23.03834715596
x19 = 85.8701998749271
x20 = 23.0383466153052
x21 = -39.7935077743833
x22 = 85.8701992984148
x23 = 35.6047169774415
x24 = 85.8702000925056
x25 = 85.8701982346336
x26 = 35.6047161176153
x27 = -14.6607647155968
x28 = -2.09439605550469
x29 = 98.4365689199204
x30 = -64.9262470331659
x31 = -52.3598771805551
x32 = 98.43656919482
x33 = -102.62536028072
x34 = -2.09439583476337
x35 = -14.6607662345448
x36 = 73.3038280578717
x37 = -77.4926165756353
x38 = 23.0383470764171
x39 = -64.9262471464508
x40 = 60.7374589963463
x41 = 48.1710882423115
x42 = 10.4719746104875
x43 = 48.1710865184453
x44 = -39.7935069083145
x45 = 85.8701991965965
x46 = 60.737457714636
x47 = -39.7935077601026
x48 = -52.3598785690371
x49 = 111.002942681265
x50 = -77.4926189336574
x51 = 98.4365697030496
x52 = 98.436570533657
x53 = -102.625359021164
x54 = -2.0943942172414
x55 = -52.3598781963485
x56 = -64.9262499512998
x57 = -14.6607661052103
x58 = 73.303829589636
x59 = 60.7374585796681
x60 = 48.1710874370125
x61 = -90.0589893487749
x62 = 35.6047157273075
x63 = -77.4926180924376
x64 = -27.2271372977792
x65 = -77.4926177917317
x66 = -52.3598773400373
x67 = 1769.76386273384
x68 = -90.0589901473458
x69 = -90.0589885272218
x70 = -90.0589886626866
x71 = -64.9262485797698
x72 = 73.3038280805586
x73 = 60.7374578443567
x74 = 60.7374570093553
x75 = 10.4719753845204
x76 = -2.09439502388267
x77 = -14.6607667421833
x78 = -77.4926193207108
x79 = 48.1710881638467
x80 = 85.8701984779953
x81 = -39.7935066904634
x82 = 73.303831094048
x83 = 48.1710865367497
x84 = 35.6047176963482
x85 = -64.9262477000829
x86 = -52.3598766114835
x87 = 73.3038289583715
x88 = -14.6607632383843
x89 = -27.2271366022125
x90 = -39.7935061477395
x90 = -39.7935061477395