# Math Help - Intersection of two functions

1. ## Intersection of two functions

y=cos(x) intersects y=x once,

find A such that y=Acos(x) intersects y=x exactly twice.

2. Hello gwtkof

Welcome to Math Help Forum!
Originally Posted by gwtkof
y=cos(x) intersects y=x once,

find A such that y=Acos(x) intersects y=x exactly twice.
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 $x=0$ and $x=-\pi$.

Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when $x \approx -2.8$, with $A\approx 2.972$.

PS
A more analytical approach is to say that the gradient of $A\cos x$ at the point when it touches the line will be $\displaystyle 1$. So if you solve the equations
$A\cos x=x$ and $-A\sin x = 1$
simultaneously, you'll end up with
$\tan x = -\dfrac1x$
which will again require a numerical method.

This gives the same approximate values as I stated above.

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 $x=0$ and $x=-\pi$.

Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when $x \approx -2.8$, with $A\approx 2.972$.

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.

4. Thanks guys that helped alot.