y=cos(x) intersects y=x once,
find A such that y=Acos(x) intersects y=x exactly twice.
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The second point of intersection will be when the line is a tangent to the curve between and .
Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when , with .
A more analytical approach is to say that the gradient of at the point when it touches the line will be . So if you solve the equations
andsimultaneously, you'll end up with
which will again require a numerical method.
This gives the same approximate values as I stated above.
cos(a) = -sin(a)a
a = -cot(a)
Then we can approximate a accordingly.
Note: When I made this post it was just in response to the quoted portion, as the PS hadn't been posted yet.