Problem
ALG-B2-M08-P008 Unit circle
#8
★★★☆☆ Level 3 of 5
Let \(x,y\ge0\), \(x^2+y^2=1\). Prove \(xy\le\frac12\).
Hint. You may set \(x=\cos t\), \(y=\sin t\).
For \(x,y\ge0\), there is \(t\in[0,\pi/2]\) such that \(x=\cos t\), \(y=\sin t\). Then \(xy=\frac12\sin2t\le\frac12\).
Teaching goal: recognize the appropriate substitution and check the domain.