Mister Exam
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Identical expressions
- sin(x/ two -pi/ six)= one
- sinus of (x divide by 2 minus Pi divide by 6) equally 1
- sinus of (x divide by two minus Pi divide by six) equally one
- sinx/2-pi/6=1
- sin(x divide by 2-pi divide by 6)=1
-
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
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
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \pi$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
$$\sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
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
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \pi$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
Sum and product of roots
[src]
sum
4*pi ---- 3
$$\frac{4 \pi}{3}$$
=
4*pi ---- 3
$$\frac{4 \pi}{3}$$
product
4*pi ---- 3
$$\frac{4 \pi}{3}$$
=
4*pi ---- 3
$$\frac{4 \pi}{3}$$
4*pi/3
Numerical answer
[src]
x1 = 41.8879013215942
x2 = -20.9439529196986
x3 = -8.37758103938507
x4 = -96.3421757169514
x5 = 4.18878718308832
x6 = 16.7551605561972
x7 = -46.0766915273204
x8 = -46.0766922582736
x9 = 92.1533836583861
x10 = 117.286125997353
x11 = -71.2094334733043
x12 = -46.0766929910768
x13 = -108.908545251863
x14 = -83.7758033036064
x15 = -46.0766921859055
x16 = 16.7551614225443
x17 = 4.1887893811571
x18 = -96.3421744988603
x19 = 79.5870128757297
x20 = -20.9439514218092
x21 = -96.3421753532747
x22 = -20.9439499803901
x23 = 29.3215304083335
x24 = 4.18879026164399
x25 = -8.37758006086171
x26 = 41.8879029376616
x27 = 67.0206443070063
x28 = 29.3215309779553
x29 = -8.37758142094941
x30 = 41.8879027415148
x31 = 29.3215309229917
x32 = -83.7758040750918
x33 = -96.3421743015664
x34 = -8.37757945753306
x35 = -71.2094328918094
x36 = 41.8879021375084
x37 = 54.4542720646
x38 = -46.0766913722027
x39 = -83.7758048986743
x40 = 117.286127281896
x41 = -83.7758049298708
x42 = 41.8879010894376
x43 = -83.7758031843618
x44 = 79.5870148501595
x45 = 92.1533836923941
x46 = -71.2094344512338
x47 = 67.0206422724735
x48 = 92.153385319565
x49 = -58.6430631914826
x50 = 4.18879100807841
x51 = 16.7551607364294
x52 = 4.18879110042936
x53 = 54.4542725437353
x54 = 67.0206437729541
x55 = 79.5870141742956
x56 = 54.4542733771966
x57 = 29.3215324370619
x58 = -33.5103206450548
x59 = -20.943949995193
x60 = -58.6430618683391
x61 = 92.1533846227311
x62 = 67.0206428941909
x63 = 29.3215318002581
x64 = -71.2094337605937
x65 = -33.5103221932076
x66 = -33.5103209358512
x67 = -46.0766932001855
x68 = -96.3421737655023
x69 = 582.241835728491
x70 = -33.5103225441362
x71 = -20.9439520320165
x72 = 67.0206411453713
x73 = -8.37758018127151
x74 = -58.6430605474131
x75 = -58.643062516056
x76 = 29.3215339606922
x77 = 54.4542717651733
x78 = -58.6430633920505
x79 = 67.0206441372174
x80 = 79.5870141361055
x81 = -20.9439505423738
x82 = -598.997000629676
x83 = 79.5870132746521
x84 = 104.719754954188
x85 = -58.643063892579
x86 = 16.7551598557508
x87 = -71.2094324512579
x88 = 4.18878937894334
x89 = 92.1533853838349
x90 = 54.4542736476806
x91 = 16.7551618472868
x92 = -33.5103217746072
x92 = -33.5103217746072
The graph