Mister Exam
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How to use it?
- Equation:
-
Equation x*8=25+15
-
Equation cos(x)=5
- Equation (x-4)^4-4*(x-4)^2-21=0
-
Equation ((1/5))^(x)^(2)*5^(2*x+2)=(1/5)^(6)
- Express {x} in terms of y where:
- 15*x-5*y=-5
- 15*x+7*y=2
- 12*x-20*y=10
- -11*x-1*y=15
- Canonical form:
- cos(x+(pi/6))
-
Identical expressions
- cos(x+(pi/ six))= one
- co sinus of e of (x plus ( Pi divide by 6)) equally 1
- co sinus of e of (x plus ( Pi divide by six)) equally one
- cosx+pi/6=1
- cos(x+(pi divide by 6))=1
-
Similar expressions
- cos(x-(pi/6))=1
- cos((x+pi)/6)=sqrt(3)/2
- cos(x+pi/6)=sqrt(1)
cos(x+(pi/6))=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(x + \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{6} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x + \frac{\pi}{6} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$x + \frac{\pi}{6} = \pi n$$
$$x + \frac{\pi}{6} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n - \frac{\pi}{6}$$
$$x = \pi n - \frac{7 \pi}{6}$$
$$\cos{\left(x + \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{6} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x + \frac{\pi}{6} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$x + \frac{\pi}{6} = \pi n$$
$$x + \frac{\pi}{6} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n - \frac{\pi}{6}$$
$$x = \pi n - \frac{7 \pi}{6}$$
Rapid solution
[src]
-pi
x1 = ----
6
$$x_{1} = - \frac{\pi}{6}$$
11*pi
x2 = -----
6
$$x_{2} = \frac{11 \pi}{6}$$
x2 = 11*pi/6
Sum and product of roots
[src]
sum
pi 11*pi - -- + ----- 6 6
$$- \frac{\pi}{6} + \frac{11 \pi}{6}$$
=
5*pi ---- 3
$$\frac{5 \pi}{3}$$
product
-pi 11*pi ----*----- 6 6
$$- \frac{\pi}{6} \frac{11 \pi}{6}$$
=
2 -11*pi ------- 36
$$- \frac{11 \pi^{2}}{36}$$
-11*pi^2/36
Numerical answer
[src]
x1 = 81.1578106196128
x2 = 56.0250685169442
x3 = 49.7418838493326
x4 = -13.0899697075722
x5 = 5.75958669034246
x6 = 68.5914394200902
x7 = -19.3731539653969
x8 = 156.556034516879
x9 = 30.8923274298613
x10 = -25.65633950106
x11 = 93.7241803230185
x12 = 56.0250690081558
x13 = -69.638637223832
x14 = -69.6386366485818
x15 = -13.0899698146708
x16 = 12.0427722166701
x17 = 81.1578098139843
x18 = -75.9218220581853
x19 = -264.417382419922
x20 = 68.5914401180933
x21 = -88.4881928048828
x22 = -57.0722660751833
x23 = -38.2227106630569
x24 = -75.9218228126903
x25 = -19.373154896527
x26 = -63.3554513363679
x27 = 62.3082540196632
x28 = -6.80678449725847
x29 = 100.007365661841
x30 = 30.8923274495049
x31 = -94.7713788080582
x32 = 62.308254257502
x33 = 12.0427713725902
x34 = -31.9395254617805
x35 = 12.0427717615073
x36 = -63.3554520547243
x37 = 62.3082547928689
x38 = 74.8746253871791
x39 = -50.7890816527148
x40 = -38.2227104687978
x41 = -13.0899689209265
x42 = 24.6091429668081
x43 = -44.5058967203222
x44 = 18.3259568926578
x45 = 37.1755126583087
x46 = 68.5914401276649
x47 = -94.7713780045959
x48 = 5.75958726070503
x49 = 37.1755134637125
x50 = 93.7241806097162
x51 = -25.6563402322524
x52 = -31.9395249202358
x53 = -0.523598489731523
x54 = 24.6091422617441
x55 = 74.8746246066628
x56 = -25.6563400637071
x57 = -44.5058964115732
x58 = 56.0250689522333
x59 = -57.0722665843695
x60 = -31.939525771185
x61 = -57.0722668645673
x62 = 18.3259570967149
x63 = -82.2050082777704
x64 = 43.4586986677198
x65 = -63.3554510011992
x66 = 301.069297155331
x67 = 30.8923282331952
x68 = 43.4586978952293
x69 = -82.2050076281063
x70 = 100.007366129962
x71 = -50.7890808483935
x72 = -38.2227111279653
x73 = -6.80678369229017
x74 = -75.9218229143423
x75 = -19.3731541846809
x76 = 49.7418835245054
x77 = -82.2050077643523
x78 = -44.5058956472658
x79 = 24.6091430772094
x80 = -69.6386373548543
x81 = 81.1578093180369
x82 = -0.523599258250701
x83 = 49.7418831724391
x84 = 18.3259576460966
x85 = 87.4409958250878
x86 = -88.4881935647341
x87 = 56.0250693524419
x88 = 5.75958602210796
x89 = 87.4409950488631
x90 = 93.7241810081872
x91 = -101.054563229573
x92 = 5.75958643339931
x93 = -88.4881942807791
x93 = -88.4881942807791
The graph