Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^4=(4x-5)^2
-
Equation (x+5)^3=25*(x+5)
-
Equation x^3+2x^2-x-2=0
-
Equation x^2=12
- Express {x} in terms of y where:
- -19*x-20*y=-4
- 13*x-18*y=-17
- 7*x+6*y=20
- 5*x+12*y=18
- Integral of d{x}:
-
cos(3x)
-
1/2
- Derivative of:
-
cos(3x)
-
1/2
- Graphing y =:
-
cos(3x)
- Sum of series:
-
1/2
-
Identical expressions
- cos(3x)= one / two
- co sinus of e of (3x) equally 1 divide by 2
- co sinus of e of (3x) equally one divide by two
- cos3x=1/2
- cos(3x)=1 divide by 2
-
Similar expressions
- (x-1)/(x+2)-(2x-1)/(2x+1)=0
- 1+cos2x+2cosxcos3x=0
- 1+tan(x)+tan(x)^(2)+tan(x)^(3)=sinx+cosx/cos^3x
- sqrt(3)/2*cos(x)-1/2*sin(x)=cos3x
- 27cos^3x–8=0
cos(3x)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(3 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$3 x = \pi n + \frac{\pi}{3}$$
$$3 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{3} - \frac{2 \pi}{9}$$
$$\cos{\left(3 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$3 x = \pi n + \frac{\pi}{3}$$
$$3 x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{3} - \frac{2 \pi}{9}$$
Rapid solution
[src]
pi
x1 = --
9
$$x_{1} = \frac{\pi}{9}$$
5*pi
x2 = ----
9
$$x_{2} = \frac{5 \pi}{9}$$
x2 = 5*pi/9
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 9 9
$$\frac{\pi}{9} + \frac{5 \pi}{9}$$
=
2*pi ---- 3
$$\frac{2 \pi}{3}$$
product
pi 5*pi --*---- 9 9
$$\frac{\pi}{9} \frac{5 \pi}{9}$$
=
2 5*pi ----- 81
$$\frac{5 \pi^{2}}{81}$$
5*pi^2/81
Numerical answer
[src]
x1 = -73.6528944341607
x2 = -1.74532925199433
x3 = -68.7659725285766
x4 = -10.1229096615671
x5 = -23.387411976724
x6 = -58.2939970166106
x7 = -71.5584993317675
x8 = -93.8987137572949
x9 = -5.93411945678072
x10 = 84.1248699461267
x11 = -54.1052068118242
x12 = 40.1425727958696
x13 = -45.7276264022514
x14 = -31.7649923862968
x15 = 5.93411945678072
x16 = -49.9164166070378
x17 = 23.387411976724
x18 = 38.0481776934764
x19 = 18.5004900711399
x20 = 73.6528944341607
x21 = 42.2369678982628
x22 = -82.0304748437335
x23 = -38.0481776934764
x24 = -40.1425727958696
x25 = 75.7472895365539
x26 = 35.9537825910832
x27 = -14.3116998663535
x28 = -95.9931088596881
x29 = 52.010811709431
x30 = 98.0875039620813
x31 = 44.331363000656
x32 = -25.4818070791172
x33 = -104.370689269261
x34 = -35.9537825910832
x35 = 20.5948851735331
x36 = 70.8603676309698
x37 = 54.1052068118242
x38 = 10.1229096615671
x39 = -12.915436464758
x40 = 88.3136601509131
x41 = 93.8987137572949
x42 = -2150248.78004967
x43 = 8.02851455917392
x44 = -87.6155284501153
x45 = -79.9360797413403
x46 = -12.2173047639603
x47 = -98.0875039620813
x48 = -69.4641042293743
x49 = -52.010811709431
x50 = -67.3697091269811
x51 = -33.85938748869
x52 = -84.1248699461267
x53 = -77.8416846389471
x54 = 31.7649923862968
x55 = -8.02851455917392
x56 = -43.6332312998582
x57 = 12.2173047639603
x58 = 33.85938748869
x59 = -91.8043186549017
x60 = -89.7099235525085
x61 = 29.6705972839036
x62 = 86.2192650485199
x63 = -27.5762021815104
x64 = 14.3116998663535
x65 = 56.1996019142174
x66 = -75.0491578357562
x67 = 60.3883921190038
x68 = 100.181899064475
x69 = -47.8220215046446
x70 = -60.3883921190038
x71 = 26.8780704807127
x72 = 77.8416846389471
x73 = -16.4060949687467
x74 = 90.4080552533063
x75 = 58.2939970166106
x76 = -75.7472895365539
x77 = -29.6705972839036
x78 = 46.4257581030492
x79 = 62.482787221397
x80 = 16.4060949687467
x81 = 95.9931088596881
x82 = 47.8220215046446
x83 = 0.349065850398866
x84 = 3.83972435438753
x85 = 64.5771823237902
x86 = -100.181899064475
x87 = -20.5948851735331
x88 = -3.83972435438753
x89 = 79.9360797413403
x90 = 82.0304748437335
x91 = -56.1996019142174
x92 = 49.9164166070378
x92 = 49.9164166070378
The graph