# Answer to Question #18415 in Calculus for freesia

Question #18415

Find the point on the curve y = cos (x) closest to the point (1,1)

Expert's answer

To the first we have to find function of distance:d=Sqrt((x-x0)^2+(y-y0)^2)

For our function: d(x)=Sqrt((x-1)^2+(cos(x)-1)^2)

Let's derive it:((x-1)-sin(x)(cos(x)-1))/Sqrt((x-1)^2+(cos(x)-1)^2))

It will be zero (minimum) at point when next equality istrue: x-1=sinx(cosx-1)

It's a transcendent equation that can be solved onlynumerical: x=0.8

So, cos(x)=0.69.

It's a point (0.8,0.69)

For our function: d(x)=Sqrt((x-1)^2+(cos(x)-1)^2)

Let's derive it:((x-1)-sin(x)(cos(x)-1))/Sqrt((x-1)^2+(cos(x)-1)^2))

It will be zero (minimum) at point when next equality istrue: x-1=sinx(cosx-1)

It's a transcendent equation that can be solved onlynumerical: x=0.8

So, cos(x)=0.69.

It's a point (0.8,0.69)

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