What is the angle of intersection of $$[|\sin x| + |\cos x|]$$ And the curve $$ x^2 + y^2 = 5 $$ where $[n]$ denotes greatest integer function.
This is a homework question. I have tried to find the intersection of these two curves but i am unable to do so. In the solution booklet provided , the first curve has been changed to $y = 1 $ without any explanation. Can anyone please explain me this ?

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    $\begingroup$ For $0\le x\le\dfrac\pi2,|\sin x|+|\cos x|=\sin x+\cos x=\sqrt2\sin\left(x+\dfrac\pi4\right)$ and $\dfrac\pi4\le x+\dfrac\pi4\le\dfrac\pi2+\dfrac\pi4$ $\implies\sqrt2\sin\left(x+\dfrac\pi4\right)\ge1$ $\implies[|\sin x|+|\cos x|]=1$ for $0\le x\le\dfrac\pi2$ $\endgroup$ – lab bhattacharjee Jul 19 '15 at 9:09
  • $\begingroup$ @labbhattacharjee Ok. Thanks. Post this as an answer and i will mark it as correct $\endgroup$ – user250085 Jul 19 '15 at 9:12

1) From the above hint it is proved that [|sinx+cosx|]=1...(eq i)

Again,if we find the local extremum for the above equation(eq i) [0<=x<=(pi/2)], we get the maximum value as (root 2)=1.414(approx), minimum value=1.

So,in this way we can also get that [|sinx+cosx|]=1.

Thus y=[|sinx|+|cosx|]=1

2) Now, find the point of intersection between y=1 & x^2+y^2=5

3) Since one of the curve is a circle, find the equation of the tangent to the circle (at the point of intersection of the straight line & the circle).

4) Find the slopes of the tangent & the straight line.

5) Now find the angle between them by using the equation [|(a-b)/(1+ab)|] (where a & b are the slopes of the tangent & straight line respectively)


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