cos(x)-2x=1

In the Internet found nothing on this equation. Please tell me method) (Except graphics)

In the Internet found nothing on this equation. Please tell me method) (Except graphics)

asked April 7th 20 at 15:47

3 answers

answered on April 7th 20 at 15:49

What's wrong with the graphical method? It clearly shows that the equation cos(x) = 1+2x is only one solution, x = 0.

answered on April 7th 20 at 15:51

https://www.wolframalpha.com/ to help you

answered on April 7th 20 at 15:53

cos(x) = 2x+1.

If the roots of this equation is, -1<= 2x+1 <=1 due to the properties of cos(x).

I.e. -1<=x<=0. Note that on this interval the function y=2x+1 is continuous and monotonically increases without any singular points.

Now consider on the same stretch of the function y = cos(x). In this case, substituting in cos(x) -1<=x<=0, you must multiply them by Pi. I.e. the function y = cos(x) need to be considered on the interval [-Pi,0]. Note that the function y = cos(x) on this interval is also continuous and monotonically increasing without any singular points.

All of this means that these two functions only intersect at one point. And you can just pick up from the segment -1<=x<=0. This is the only point x = 0.

Well, not a trivial task. At least the other approaches do not see.

If the roots of this equation is, -1<= 2x+1 <=1 due to the properties of cos(x).

I.e. -1<=x<=0. Note that on this interval the function y=2x+1 is continuous and monotonically increases without any singular points.

Now consider on the same stretch of the function y = cos(x). In this case, substituting in cos(x) -1<=x<=0, you must multiply them by Pi. I.e. the function y = cos(x) need to be considered on the interval [-Pi,0]. Note that the function y = cos(x) on this interval is also continuous and monotonically increasing without any singular points.

All of this means that these two functions only intersect at one point. And you can just pick up from the segment -1<=x<=0. This is the only point x = 0.

Well, not a trivial task. At least the other approaches do not see.

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