Mister Exam
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How to use it?
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- two *cos(x/ two)+ one = zero
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Similar expressions
- sqrt(2)*cos(x/2)+1=0
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2*cos(x/2)+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \cos{\left(\frac{x}{2} \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(\frac{x}{2} \right)} = -1$$
The equation is transformed to
$$\cos{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
$$2 \cos{\left(\frac{x}{2} \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$2 \cos{\left(\frac{x}{2} \right)} = -1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
Rapid solution
[src]
4*pi
x1 = ----
3
$$x_{1} = \frac{4 \pi}{3}$$
8*pi
x2 = ----
3
$$x_{2} = \frac{8 \pi}{3}$$
x2 = 8*pi/3
Sum and product of roots
[src]
sum
4*pi 8*pi ---- + ---- 3 3
$$\frac{4 \pi}{3} + \frac{8 \pi}{3}$$
=
4*pi
$$4 \pi$$
product
4*pi 8*pi ----*---- 3 3
$$\frac{4 \pi}{3} \frac{8 \pi}{3}$$
=
2 32*pi ------ 9
$$\frac{32 \pi^{2}}{9}$$
32*pi^2/9
Numerical answer
[src]
x1 = -67.0206432765823
x2 = -192.684349420174
x3 = -33.5103216382911
x4 = 8.37758040957278
x5 = -41.8879020478639
x6 = -79.5870138909414
x7 = 92.1533845053006
x8 = -4.18879020478639
x9 = -20.943951023932
x10 = 41.8879020478639
x11 = -46.0766922526503
x12 = 67.0206432765823
x13 = 54.4542726622231
x14 = -117.286125734019
x15 = -83.7758040957278
x16 = 96.342174710087
x17 = -71.2094334813686
x18 = -8.37758040957278
x19 = 71.2094334813686
x20 = 16.7551608191456
x21 = -96.342174710087
x22 = -58.6430628670095
x23 = 29.3215314335047
x24 = 33.5103216382911
x25 = -29.3215314335047
x26 = 83.7758040957278
x27 = 58.6430628670095
x28 = -92.1533845053006
x29 = 79.5870138909414
x30 = 46.0766922526503
x31 = -16.7551608191456
x32 = 20.943951023932
x33 = 4.18879020478639
x34 = -54.4542726622231
x34 = -54.4542726622231
The graph