Question

Determinați valoarea maximă a expresiei sinX/2 +cosx/2

Answer 1

$sin\dfrac{x}2+cos\dfrac{x}2=sin\dfrac{x}2+sin\left(\dfrac{\pi}2-\dfrac{x}2\right)=\\\\=2\cdot sin\dfrac{\dfrac{x}2+\dfrac{\pi}2-\dfrac{x}2}2\cdot cos\dfrac{\dfrac{x}2-\dfrac{\pi}2+\dfrac{x}2}2=\\\\=2\cdot sin\dfrac{\pi}4\cdot cos\left(\dfrac{x}2-\dfrac{\pi}4\right)=\sqrt2\cdot cos\left(\dfrac{x}2-\dfrac{\pi}4\right).$

Valoarea maximă a lui cos este 1, deci valoare maximă a expresiei este √2.

Green eyes.