y=cos(x) intersects y=x once,
find A such that y=Acos(x) intersects y=x exactly twice.
Hello gwtkof
Welcome to Math Help Forum!I think you will need to use a numerical/graphical method to solve this.
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 .
Grandad
PS
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.
I think there's a more straightforward way. Let a be the value that you approximated as -2.8. The slope of the tangent of y=cos(x) at this point is -sin(a). The line going through the origin and the point of tangency is y=-sin(a)x. The point of intersection of the line y=-sin(a)x and the curve y=cos(x) gives us the relation
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.