Mister Exam
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How to use it?
- Equation:
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Equation sin(x)=1/5
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Equation x^2+11=0
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Equation x^2+5x+6=0
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Equation -x/7+x=15/7
- Express {x} in terms of y where:
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- -4*x-8*y=-20
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-
Identical expressions
- two *sin(x-(pi/ four))= one
- 2 multiply by sinus of (x minus ( Pi divide by 4)) equally 1
- two multiply by sinus of (x minus ( Pi divide by four)) equally one
- 2sin(x-(pi/4))=1
- 2sinx-pi/4=1
- 2*sin(x-(pi divide by 4))=1
-
Similar expressions
- 2*sin(x+(pi/4))=1
- sqrt(2)*sin(x-pi/4)=-1
- 2*sin(x-pi/4)-sqrt(2)=0
2*sin(x-(pi/4))=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ pi\
2*sin|x - --| = 1
\ 4 /
$$2 \sin{\left(x - \frac{\pi}{4} \right)} = 1$$
Detail solution
Given the equation
$$2 \sin{\left(x - \frac{\pi}{4} \right)} = 1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{4} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x + \frac{\pi}{4} = \pi n + \frac{2 \pi}{3}$$
$$x + \frac{\pi}{4} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n + \frac{5 \pi}{12}$$
$$x = \pi n - \frac{7 \pi}{12}$$
$$2 \sin{\left(x - \frac{\pi}{4} \right)} = 1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{4} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x + \frac{\pi}{4} = \pi n + \frac{2 \pi}{3}$$
$$x + \frac{\pi}{4} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n + \frac{5 \pi}{12}$$
$$x = \pi n - \frac{7 \pi}{12}$$
Sum and product of roots
[src]
sum
5*pi 13*pi ---- + ----- 12 12
$$\frac{5 \pi}{12} + \frac{13 \pi}{12}$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
product
5*pi 13*pi ----*----- 12 12
$$\frac{5 \pi}{12} \frac{13 \pi}{12}$$
=
2 65*pi ------ 144
$$\frac{65 \pi^{2}}{144}$$
65*pi^2/144
Rapid solution
[src]
5*pi
x1 = ----
12
$$x_{1} = \frac{5 \pi}{12}$$
13*pi
x2 = -----
12
$$x_{2} = \frac{13 \pi}{12}$$
x2 = 13*pi/12
Numerical answer
[src]
x1 = 20.1585528605345
x2 = 95.5567765466895
x3 = 13.8753675533549
x4 = 28.5361332701073
x5 = 34.8193185772869
x6 = 41.1025038844665
x7 = -67.8060414399797
x8 = 15.9697626557481
x9 = -97.1275728734844
x10 = -65.7116463375865
x11 = 47.3856891916461
x12 = -30.1069295969022
x13 = 85.0848010347236
x14 = -80.3724120543389
x15 = 59.9520598060052
x16 = 22.2529479629277
x17 = 9.68657734856853
x18 = -40.5789051088682
x19 = 72.5184304203644
x20 = -34.2957198016886
x21 = 26.4417381677141
x22 = -46.8620904160477
x23 = 45.2912940892529
x24 = 110.217542263442
x25 = 91.3679863419031
x26 = -2.87979326579064
x27 = -36.3901149040818
x28 = -74.0892267471593
x29 = 39.0081087820733
x30 = 1.30899693899575
x31 = -53.1452757232273
x32 = -92.9387826686981
x33 = -21.7293491873294
x34 = 342.695398629087
x35 = 7.59218224617533
x36 = -61.5228561328001
x37 = -28.012534494509
x38 = -17.540558982543
x39 = 70.4240353179712
x40 = -84.5612022591253
x41 = -23.8237442897226
x42 = 57.857664703612
x43 = -99.2219679758776
x44 = 3.40339204138894
x45 = -15.4461638801498
x46 = 78.801615727544
x47 = 53.6688744988256
x48 = -59.4284610304069
x49 = 32.7249234748937
x50 = 103.934356956262
x51 = -306.567083112804
x52 = -71.9948316447661
x53 = 97.6511716490827
x54 = 76.7072206251508
x55 = -90.8443875663049
x56 = -4.97418836818384
x57 = 51.5744793964324
x58 = -48.9564855184409
x59 = -9.16297857297023
x60 = -11.2573736753634
x61 = -55.2396708256205
x62 = -86.6555973615185
x63 = 89.27359123951
x64 = 66.2352451131848
x65 = 82.9904059323304
x66 = -78.2780169519457
x67 = 64.1408500107916
x68 = -42.6733002112614
x68 = -42.6733002112614
The graph