Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2=49
-
Equation cos(pi*(x-1)/3)=1/2
-
Equation 3*x^2-3=0
-
Equation 5(7x+3)-12=4(9x-6)
- Express {x} in terms of y where:
- 18*x-14*y=-12
- 4*x-18*y=-14
- -14*x-1*y=3
- 10*x-18*y=6
- Sum of series:
-
-1
- Derivative of:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- sin(x/ three +pi/ eighteen)=- one
- sinus of (x divide by 3 plus Pi divide by 18) equally minus 1
- sinus of (x divide by three plus Pi divide by eighteen) equally minus one
- sinx/3+pi/18=-1
- sin(x divide by 3+pi divide by 18)=-1
-
Similar expressions
- sin(x/3+pi/18)=+1
- sin(x/3-pi/18)=-1
sin(x/3+pi/18)=-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ sin|- + --| = -1 \3 18/
$$\sin{\left(\frac{x}{3} + \frac{\pi}{18} \right)} = -1$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{3} + \frac{\pi}{18} \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n - \frac{\pi}{2}$$
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n + \frac{3 \pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{18}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n - \frac{5 \pi}{9}$$
$$\frac{x}{3} = 2 \pi n + \frac{13 \pi}{9}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n - \frac{5 \pi}{3}$$
$$x_{2} = 6 \pi n + \frac{13 \pi}{3}$$
$$\sin{\left(\frac{x}{3} + \frac{\pi}{18} \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n - \frac{\pi}{2}$$
$$\frac{x}{3} + \frac{\pi}{18} = 2 \pi n + \frac{3 \pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{18}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n - \frac{5 \pi}{9}$$
$$\frac{x}{3} = 2 \pi n + \frac{13 \pi}{9}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n - \frac{5 \pi}{3}$$
$$x_{2} = 6 \pi n + \frac{13 \pi}{3}$$
Rapid solution
[src]
-5*pi
x1 = -----
3
$$x_{1} = - \frac{5 \pi}{3}$$
13*pi
x2 = -----
3
$$x_{2} = \frac{13 \pi}{3}$$
x2 = 13*pi/3
Sum and product of roots
[src]
sum
5*pi 13*pi - ---- + ----- 3 3
$$- \frac{5 \pi}{3} + \frac{13 \pi}{3}$$
=
8*pi ---- 3
$$\frac{8 \pi}{3}$$
product
-5*pi 13*pi -----*----- 3 3
$$- \frac{5 \pi}{3} \frac{13 \pi}{3}$$
=
2 -65*pi ------- 9
$$- \frac{65 \pi^{2}}{9}$$
-65*pi^2/9
Numerical answer
[src]
x1 = -99.4837702451178
x2 = 70.1622346562496
x3 = 70.1622373962342
x4 = -61.7846565105811
x5 = -99.4837659692826
x6 = -80.6342110919223
x7 = -5.23598846337339
x8 = -5.23598802448246
x9 = 13.6135690415039
x10 = 70.1622360280877
x11 = 51.3126794910504
x12 = 51.3126787396147
x13 = -24.0855450675216
x14 = 32.4631213181673
x15 = 89.0117914967097
x16 = -80.6342129617026
x17 = 32.4631255231335
x18 = 13.6135692374962
x19 = 51.3126803832776
x20 = 70.1622350570105
x21 = 13.6135666870516
x22 = -99.4837668751494
x23 = 89.0117906740962
x24 = 13.6135702642396
x25 = 32.463124832646
x26 = -42.9350997355404
x27 = -42.935098349142
x28 = -80.6342127262422
x29 = -99.4837666401773
x30 = -80.6342106151371
x31 = -99.4837659119862
x32 = 13.6135675174231
x33 = 51.3126815568832
x34 = 89.0117921626959
x35 = -5.23598624918
x36 = -24.085542714959
x37 = 51.3126811818992
x38 = 32.463123030893
x39 = -80.6342109289082
x40 = -24.0855444513601
x41 = -42.9350980578904
x42 = -42.9351007935127
x43 = 70.1622367973115
x44 = 32.4631225777541
x45 = -80.6342098945046
x46 = -80.6342102861259
x47 = 6554.40947713682
x48 = -42.9350997871399
x49 = 13.6135668155749
x50 = -24.0855436047653
x51 = 89.0117915921671
x52 = 89.0117923619355
x53 = 51.3126784833101
x54 = -24.0855421731002
x55 = 32.4631254934077
x56 = 51.3126806260284
x57 = -80.6342104205738
x58 = 107.861351429411
x59 = -5.23598885133824
x60 = -24.0855398847336
x61 = -61.784656763105
x62 = -99.4837675134227
x63 = 13.6135697015054
x64 = -24.0855425159028
x65 = 70.1622372452464
x66 = 32.463124570374
x67 = 32.4631239639637
x68 = -5.23598929110112
x69 = -80.6342119830858
x70 = 89.0117903107836
x71 = 89.0117931171951
x72 = 89.0117914689468
x73 = -61.7846554920137
x74 = 51.3126804228802
x75 = -61.7846564145052
x76 = -24.0855450966611
x77 = -5.23598713749819
x78 = -5.23598913593662
x79 = -61.7846540640803
x80 = 32.4631230991761
x81 = 89.0117933890366
x82 = -99.4837696384033
x83 = -61.7846569954863
x84 = -42.9350991124378
x85 = -5.23598642220758
x86 = -61.7846546744869
x87 = 32.4631208538596
x88 = 70.1622344640687
x89 = -42.9351011391729
x90 = -42.9351000059199
x91 = 13.6135683990585
x92 = -61.7846541796602
x92 = -61.7846541796602