Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation (x+2)^2=(1-x)^2
-
Equation 2*cos(x)+1=0
-
Equation (-5*x-3)*(2*x-1)=0
-
Equation x^2-8x+15=0
- Express {x} in terms of y where:
- 7*x+13*y=-13
- 7*x+19*y=12
- 1*x+20*y=-17
- 14*x-16*y=-3
- Graphing y =:
- 2*cos(x)+1
-
Identical expressions
- two *cos(x)+ one = zero
- 2 multiply by co sinus of e of (x) plus 1 equally 0
- two multiply by co sinus of e of (x) plus one equally zero
- 2cos(x)+1=0
- 2cosx+1=0
- 2*cos(x)+1=O
-
Similar expressions
- sin(x)+sqrt((2-sqrt3)/2*(cos(x)+1))=0
- 2sin^2(x)-sqrt(3)sin(x)/2cos(x)+1=0
- 2*cos(x)-1=0
- sin(x)+sqrt(((2-(sqrt(3)))/2)*(cos(x)+1))=0
- sinx+((2-(3)^0,5)/2*(cosx+1))^0,5=0
- 2*cosx+1=0
2*cos(x)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \cos{\left(x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(x \right)} = -1$$
The equation is transformed to
$$\cos{\left(x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x = \pi n + \frac{2 \pi}{3}$$
$$x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
$$2 \cos{\left(x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$2 \cos{\left(x \right)} = -1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x = \pi n + \frac{2 \pi}{3}$$
$$x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Rapid solution
[src]
2*pi
x1 = ----
3
$$x_{1} = \frac{2 \pi}{3}$$
4*pi
x2 = ----
3
$$x_{2} = \frac{4 \pi}{3}$$
Sum and product of roots
[src]
sum
2*pi 4*pi
0 + ---- + ----
3 3
$$\left(0 + \frac{2 \pi}{3}\right) + \frac{4 \pi}{3}$$
=
2*pi
$$2 \pi$$
product
2*pi 4*pi 1*----*---- 3 3
$$\frac{4 \pi}{3} \cdot 1 \cdot \frac{2 \pi}{3}$$
=
2 8*pi ----- 9
$$\frac{8 \pi^{2}}{9}$$
8*pi^2/9
Numerical answer
[src]
x1 = -4.18879020478639
x2 = -35.6047167406843
x3 = 41.8879020478639
x4 = -90.0589894029074
x5 = -16.7551608191456
x6 = 71.2094334813686
x7 = -46.0766922526503
x8 = 73.3038285837618
x9 = -83.7758040957278
x10 = 2.0943951023932
x11 = -20.943951023932
x12 = 60.7374579694027
x13 = 4.18879020478639
x14 = 52.3598775598299
x15 = -96.342174710087
x16 = -64.9262481741891
x17 = 23.0383461263252
x18 = 29.3215314335047
x19 = -71.2094334813686
x20 = -1623.15620435473
x21 = -39.7935069454707
x22 = -8.37758040957278
x23 = 35.6047167406843
x24 = -29.3215314335047
x25 = 14.6607657167524
x26 = 8.37758040957278
x27 = 39.7935069454707
x28 = 630.412925820352
x29 = -54.4542726622231
x30 = 46.0766922526503
x31 = -2.0943951023932
x32 = -60.7374579694027
x33 = 79.5870138909414
x34 = 20.943951023932
x35 = 96.342174710087
x36 = 33.5103216382911
x37 = -33.5103216382911
x38 = -14.6607657167524
x39 = -73.3038285837618
x40 = -58.6430628670095
x41 = 117.286125734019
x42 = 98.4365698124802
x43 = 64.9262481741891
x44 = -98.4365698124802
x45 = 54.4542726622231
x46 = -10.471975511966
x47 = -85.870199198121
x48 = -92.1533845053006
x49 = 48.1710873550435
x50 = -79.5870138909414
x51 = 90.0589894029074
x52 = 16.7551608191456
x53 = 85.870199198121
x54 = 92.1533845053006
x55 = -48.1710873550435
x56 = -41.8879020478639
x57 = -27.2271363311115
x58 = -52.3598775598299
x59 = 58.6430628670095
x60 = 10.471975511966
x61 = 67.0206432765823
x62 = 83.7758040957278
x63 = -77.4926187885482
x64 = 416.784625376246
x65 = -67.0206432765823
x66 = -23.0383461263252
x67 = 27.2271363311115
x68 = 77.4926187885482
x68 = 77.4926187885482
The graph