Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -5*x=25
-
Equation 5*sin(x)+cos(x)=5
-
Equation 4*x^2+7=7+24*x
-
Equation 4x+6=2(2x+3)
- Express {x} in terms of y where:
- -11*x-1*y=12
- 6*x+4*y=15
- 3*x+19*y=4
- -12*x-11*y=19
- Sum of series:
-
-1
- Derivative of:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- cos(x-(pi/ six))=- one
- co sinus of e of (x minus ( Pi divide by 6)) equally minus 1
- co sinus of e of (x minus ( Pi divide by six)) equally minus one
- cosx-pi/6=-1
- cos(x-(pi divide by 6))=-1
-
Similar expressions
- cos(x-pi/6)=1
- cos(x+(pi/6))=-1
- cos(x-(pi/6))=+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
$$x + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Or
$$x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{2}$$
$$x + \frac{\pi}{3} = 2 \pi n + \frac{3 \pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{5 \pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
$$\cos{\left(x - \frac{\pi}{6} \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Or
$$x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{2}$$
$$x + \frac{\pi}{3} = 2 \pi n + \frac{3 \pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{5 \pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
Sum and product of roots
[src]
sum
5*pi 7*pi - ---- + ---- 6 6
$$- \frac{5 \pi}{6} + \frac{7 \pi}{6}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
-5*pi 7*pi -----*---- 6 6
$$- \frac{5 \pi}{6} \frac{7 \pi}{6}$$
=
2 -35*pi ------- 36
$$- \frac{35 \pi^{2}}{36}$$
-35*pi^2/36
Rapid solution
[src]
-5*pi
x1 = -----
6
$$x_{1} = - \frac{5 \pi}{6}$$
7*pi
x2 = ----
6
$$x_{2} = \frac{7 \pi}{6}$$
x2 = 7*pi/6
Numerical answer
[src]
x1 = -21.4675495511349
x2 = 97.9129705388194
x3 = 66.4970446206683
x4 = 60.2138596644571
x5 = -71.7330322690909
x6 = -46.6002914958496
x7 = -59.1666611578459
x8 = -96.8657738820537
x9 = 9.94837702122506
x10 = 97.9129711103689
x11 = -78.0162180479322
x12 = 47.6474879940301
x13 = 35.0811183901833
x14 = 79.0634147422311
x15 = 3.66519092286611
x16 = -15.1843644158927
x17 = -15.1843647726834
x18 = -84.2994033796035
x19 = 53.9306741489271
x20 = 53.9306733915096
x21 = -40.3171061212146
x22 = -59.1666621329751
x23 = -90.5825886498278
x24 = 66.4970450174627
x25 = 91.6297850397432
x26 = 16.2315616862384
x27 = 41.3643028125296
x28 = 53.9306739481032
x29 = 3.66519094039316
x30 = 85.3465999669243
x31 = -8.90117957011075
x32 = -2.6179935547329
x33 = -65.4498466272906
x34 = 47.6474880744246
x35 = 79.0634155455067
x36 = 35.0811182902876
x37 = -2.61799434172942
x38 = 16.2315620579542
x39 = -34.0339201083054
x40 = -96.8657730764531
x41 = 22.5147478673811
x42 = -40.317106228479
x43 = -34.0339209018893
x44 = -65.4498464335014
x45 = 16.231562520136
x46 = 41.3643036022671
x47 = -90.5825878689909
x48 = 22.5147475317097
x49 = 66.4970443621975
x50 = 35.0811175860113
x51 = -21.4675492829431
x52 = -65.4498471139063
x53 = -59.1666619301013
x54 = -84.2994026839722
x55 = -34.0339203797704
x56 = -78.0162172395008
x57 = 97.9129712746173
x58 = -84.2994031904943
x59 = -71.7330325806169
x60 = 85.3466007591799
x61 = -52.8834767261331
x62 = 47.6474887914533
x63 = 72.7802295423785
x64 = 9.94837678485442
x65 = -21.467549955294
x66 = -27.7507354478924
x67 = 60.2138588216583
x68 = 60.213859223786
x69 = 85.3466014030704
x70 = 72.7802303022061
x71 = 28.7979331490089
x72 = 9.94837624459935
x73 = 28.7979323848102
x74 = -27.7507346247054
x75 = 91.629785949731
x76 = -46.6002907118192
x77 = -78.0162175464145
x78 = 3.66519163307997
x79 = -52.8834759207986
x80 = -8.90117876525242
x81 = -40.3171055253934
x82 = 91.6297852261896
x83 = -15.1843640043429
x84 = -71.7330317700361
x85 = 22.5147472034056
x86 = -27.7507351061628
x86 = -27.7507351061628