Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^4-4*x^3-2*x^2+12*x+9=0
-
Equation x^2-5x-24=0
-
Equation 3x^2-27=0
-
Equation sin(x)-cos(x)=-1
- Express {x} in terms of y where:
- -6*x-2*y=-10
- 18*x+14*y=-11
- -5*x+6*y=19
- -10*x+9*y=-6
-
Identical expressions
- cos(x+(pi/ three))= one
- co sinus of e of (x plus ( Pi divide by 3)) equally 1
- co sinus of e of (x plus ( Pi divide by three)) equally one
- cosx+pi/3=1
- cos(x+(pi divide by 3))=1
-
Similar expressions
- 2*cos(x+pi/3)=5
- cos(x-(pi/3))=1
- cos(x+pi/3)=1
cos(x+(pi/3))=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(x + \frac{\pi}{3} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$x + \frac{\pi}{3} = \pi n$$
$$x + \frac{\pi}{3} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n - \frac{\pi}{3}$$
$$x = \pi n - \frac{4 \pi}{3}$$
$$\cos{\left(x + \frac{\pi}{3} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$x + \frac{\pi}{3} = \pi n$$
$$x + \frac{\pi}{3} = \pi n - \pi$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n - \frac{\pi}{3}$$
$$x = \pi n - \frac{4 \pi}{3}$$
Sum and product of roots
[src]
sum
pi 5*pi - -- + ---- 3 3
$$- \frac{\pi}{3} + \frac{5 \pi}{3}$$
=
4*pi ---- 3
$$\frac{4 \pi}{3}$$
product
-pi 5*pi ----*---- 3 3
$$- \frac{\pi}{3} \frac{5 \pi}{3}$$
=
2 -5*pi ------ 9
$$- \frac{5 \pi^{2}}{9}$$
-5*pi^2/9
Rapid solution
[src]
-pi
x1 = ----
3
$$x_{1} = - \frac{\pi}{3}$$
5*pi
x2 = ----
3
$$x_{2} = \frac{5 \pi}{3}$$
x2 = 5*pi/3
Numerical answer
[src]
x1 = 99.483766182202
x2 = -19.8967539141283
x3 = 30.3687289114122
x4 = 61.7846550114646
x5 = -101.578161953261
x6 = 68.067840622896
x7 = 17.8023585073476
x8 = 86.9173972167005
x9 = 42.9350992719286
x10 = -89.0117914013259
x11 = 55.5014704889569
x12 = -1.047197884948
x13 = -51.3126788801149
x14 = -7.33038234970242
x15 = 93.2005824445644
x16 = -57.5958648046158
x17 = -63.8790503300893
x18 = -45.0294942466274
x19 = 17.8023578617743
x20 = -76.4454224109362
x21 = -57.5958658923729
x22 = 86.9173964288891
x23 = 30.368728796664
x24 = -51.3126802348481
x25 = 93.2005816398396
x26 = -76.4454217322964
x27 = -95.2949766541678
x28 = -1.04719709205516
x29 = -19.8967530533671
x30 = 11.5191718122226
x31 = -57.5958654382348
x32 = 74.3510260714291
x33 = 49.2182852884197
x34 = -38.7463090241633
x35 = 5.23598732913431
x36 = -13.6135676562812
x37 = -70.162235824498
x38 = 61.7846556667327
x39 = 42.935098044129
x40 = 80.6342112394353
x41 = 17.8023583939831
x42 = 36.6519148022553
x43 = 36.6519151871741
x44 = 68.0678412242824
x45 = 24.0855440859231
x46 = 5.23598813217661
x47 = -45.029495041739
x48 = 74.3510259182365
x49 = -7.33038184400076
x50 = 24.0855429847921
x51 = 55.5014697264737
x52 = -76.4454209842472
x53 = 74.3510266393966
x54 = -89.011792198443
x55 = -57.5958654079676
x56 = 11.5191733313054
x57 = -63.8790510556062
x58 = -26.1799392876319
x59 = -51.3126795018451
x60 = -26.1799387133702
x61 = -32.4631238264292
x62 = 80.6342119540229
x63 = -63.8790501935183
x64 = 55.501469168008
x65 = -7.33038307682589
x66 = 49.21828448443
x67 = 11.5191725733813
x68 = 42.9351000623883
x69 = 24.0855432280792
x70 = 86.9173965904982
x71 = -95.2949773927983
x72 = -38.7463098260451
x73 = 36.6519140812684
x74 = 30.3687294917281
x75 = -19.8967525986102
x76 = 68.0678403709668
x77 = -82.7286061805984
x78 = -70.1622364367418
x79 = -70.1622358120646
x80 = -26.1799386522404
x81 = -82.7286069811057
x82 = -32.4631245795436
x83 = -13.6135683222534
x84 = -95.2949759363908
x85 = 80.6342122207782
x86 = -13.6135682483753
x87 = 99.4837676465256
x88 = 99.4837668797295
x89 = -32.4631253376294
x90 = 86.9173968009572
x91 = 61.784655497151
x92 = 86.9173974067754
x92 = 86.9173974067754