Mister Exam
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Identical expressions
- sin(x/ two +pi/ six)= one
- sinus of (x divide by 2 plus Pi divide by 6) equally 1
- sinus of (x divide by two plus Pi divide by six) equally one
- sinx/2+pi/6=1
- sin(x divide by 2+pi divide by 6)=1
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Similar expressions
- sin(x/2-pi/6)=1
sin(x/2+pi/6)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ sin|- + --| = 1 \2 6 /
$$\sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} = 1$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 4 \pi n + \frac{2 \pi}{3}$$
$$\sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 4 \pi n + \frac{2 \pi}{3}$$
Sum and product of roots
[src]
sum
2*pi ---- 3
$$\frac{2 \pi}{3}$$
=
2*pi ---- 3
$$\frac{2 \pi}{3}$$
product
2*pi ---- 3
$$\frac{2 \pi}{3}$$
=
2*pi ---- 3
$$\frac{2 \pi}{3}$$
2*pi/3
Numerical answer
[src]
x1 = -60.7374575765819
x2 = -98.4365688486372
x3 = 2.094395949061
x4 = 14.6607663563477
x5 = 77.4926177661927
x6 = -73.3038295357178
x7 = -23.0383464826842
x8 = 27.2271353092822
x9 = 27.2271354596038
x10 = 190.589955834988
x11 = 14.6607667367805
x12 = 77.4926182090087
x13 = -85.8701989907764
x14 = -48.171086720341
x15 = 64.9262478335924
x16 = 64.9262492069477
x17 = 14.6607647720459
x18 = -35.6047174019333
x19 = -85.8701983766599
x20 = 77.4926188658158
x21 = -48.1710872312643
x22 = 77.4926190778258
x23 = 27.2271373460175
x24 = 90.0589886195778
x25 = 27.2271367394256
x26 = -48.1710880606733
x27 = -73.3038288002941
x28 = -60.7374569584603
x29 = -48.1710845128974
x30 = -98.4365695561871
x31 = -10.4719752391043
x32 = 64.9262487094538
x33 = -60.737458455442
x34 = -23.0383456055441
x35 = -35.6047168285168
x36 = -23.0383471230098
x37 = 77.4926197655044
x38 = 2.09439593448735
x39 = -48.1710883293905
x40 = 90.0589902454253
x41 = -1166.5780726783
x42 = 64.9262456460236
x43 = -35.6047160052075
x44 = -23.0383485863509
x45 = 27.2271384496792
x46 = -98.4365704196816
x47 = 14.6607654291303
x48 = 52.3598785201906
x49 = -48.1710864501938
x50 = 90.0589892611879
x51 = -35.6047176225377
x52 = -60.7374556333531
x53 = 2.09439430780447
x54 = 52.3598774912244
x55 = 52.3598768644349
x56 = -73.3038275601839
x57 = -23.0383468971228
x58 = -60.7374589928621
x59 = 39.7935078590308
x60 = -60.7374586727562
x61 = 14.6607654977902
x62 = 39.7935070883423
x63 = -10.4719761054059
x64 = -85.870198331802
x65 = -98.4365695760279
x66 = 90.0589885077517
x67 = -23.0383450940632
x68 = 27.2271358599184
x69 = -10.4719745414195
x70 = 39.7935062524543
x71 = 52.3598783073797
x72 = 52.3598766874147
x73 = -10.4719765321465
x74 = -73.3038279576214
x75 = -10.4719753514634
x76 = 39.7935075406608
x77 = 64.9262471824245
x78 = -73.3038288193291
x79 = 102.625359080089
x80 = 39.7935059628102
x81 = -85.8702000037712
x82 = -85.8702000581254
x83 = 90.0589902282389
x84 = -35.604715770129
x85 = -23.0383455361077
x86 = 2.09439422406579
x87 = 102.625359753265
x88 = 102.625359664659
x89 = 64.9262486211609
x89 = 64.9262486211609
The graph