I tried this but could not get much far.Originally Posted by JakeD
$\displaystyle x=cosA (0<A<\frac{\pi}{2})$
$\displaystyle y=cosB (0<B<\frac{\pi}{2})$
$\displaystyle cosA + cosB \leq 1 \leq cos^2A + cos^2B$
KeepSmiling
Malay
Call this region $\displaystyle D$ and rewrite it asOriginally Posted by JakeD
$\displaystyle \begin{aligned}
0 \le x &\le 1 \\
1-x \le y &\le \sqrt{1-x^2}. \\
\end{aligned}$
The probability, which is a double integral, can then be evaluated as an iterated integral
$\displaystyle \begin{aligned}P &= \underset{D\ \ }{\iint} 1\ dxdy \\
&= \int_0^1 dx\ \int_{1-x}^{\sqrt{1-x^2}} 1\ dy \\
&= \int_0^1 \sqrt{1-x^2} - (1-x)\ dx \\
&= \sin^{-1}(1)/2 - (1 - 1/2) \\
&= \frac{\pi -2}{4}.
$
I used The Integrator to find the antiderivative of $\displaystyle \sqrt{1-x^2} .$
Great Work JakeD!Originally Posted by JakeD
I want to tell you that this problem is assigned to a student with the instructions to avoid calculus and coordinate geometry for this question(I am sorry, I should have told you this before).
You can use trigonometry and combanotrics,algebra and number theory.
Keep Smiling
Malay
What is the point of this problem? To take something very simple and make it complicated with combinatorics and number theory? Or is it to keep it simple? I think it is the latter. So I'd just go with the picture as your rough sketch and be done with it.Originally Posted by malaygoel