Problem
ALG-B2-M08-P016 Sum on the circle
#16
★★★★★ Level 5 of 5
Let \(x,y\ge0\), \(x^2+y^2=1\). Prove \(x+y\le\sqrt{2}\).
Hint. Use \(x=\cos t\), \(y=\sin t\), or Cauchy.
Let \(x=\cos t\), \(y=\sin t\). Then \(x+y=\sqrt{2}\sin(t+\pi/4)\le\sqrt{2}\).
Teaching goal: recognize the appropriate substitution and check the domain.