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Similar expressions
- 2*cos(x/2+pi/3)+1=0
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2*cos(x/2+pi/3)-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\
2*cos|- + --| - 1 = 0
\2 3 /
$$2 \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} - 1 = 0$$
Detail solution
Given the equation
$$2 \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} - 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 1$$
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n$$
$$\frac{x}{2} = \pi n - \pi$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n - 2 \pi$$
$$2 \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move -1 to right part of the equation
with the change of sign in -1
We get:
$$2 \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n$$
$$\frac{x}{2} = \pi n - \pi$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n - 2 \pi$$
Sum and product of roots
[src]
sum
8*pi ---- 3
$$\frac{8 \pi}{3}$$
=
8*pi ---- 3
$$\frac{8 \pi}{3}$$
product
8*pi 0*---- 3
$$0 \frac{8 \pi}{3}$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 20.943951023932
x2 = -37.6991118430775
x3 = -92.1533845053006
x4 = -79.5870138909414
x5 = 12.5663706143592
x6 = -54.4542726622231
x7 = -29.3215314335047
x8 = 0.0
x9 = -87.9645943005142
x10 = -75.398223686155
x11 = 62.8318530717959
x12 = 100.530964914873
x13 = 75.398223686155
x14 = 33.5103216382911
x15 = -67.0206432765823
x16 = 25.1327412287183
x17 = 58.6430628670095
x18 = -16.7551608191456
x19 = -25.1327412287183
x20 = 113.097335529233
x21 = 46.0766922526503
x22 = 87.9645943005142
x23 = -50.2654824574367
x24 = -100.530964914873
x25 = -4.18879020478639
x26 = 50.2654824574367
x27 = 1566.60753659011
x28 = -62.8318530717959
x29 = 8.37758040957278
x30 = 83.7758040957278
x31 = -12.5663706143592
x32 = 96.342174710087
x33 = 37.6991118430775
x34 = 71.2094334813686
x35 = 121.474915938805
x36 = -41.8879020478639
x36 = -41.8879020478639