Question

Pentru ce valoare a lui x

$\dfrac{cos \ x}{tg \ 30}+sin\ x$

are valoare maxima?

(30 este in grade)

Answer 1

$E = \dfrac{\cos x}{\tan 30^{\circ}}+\sin x\\ \\ \Rightarrow E= \sqrt{3}\cos x+\sin x\\ \\ \Rightarrow \dfrac{E}{2} = \dfrac{\sqrt{3}}{2}\cos x+\dfrac{1}{2}\sin x\\ \\\Rightarrow \dfrac{E}{2} = \sin(60^{\circ})\cos x+\cos(60^{\circ})\sin x\\ \\ \Rightarrow \dfrac{E}{2} = \sin(60^{\circ}+x)\\ \\ \Rightarrow E = 2\sin(60^{\circ}+x)\\ \\ -1\leq \sin(60^{\circ}+x)\leq 1 \Rightarrow -2\leq 2\sin(60^{\circ}+x)\leq 2 \Rightarrow \\ \\ \Rightarrow \sin(60^{\circ}+x)=1 \Rightarrow \sin(60^{\circ}+x)=\sin(90^{\circ})\Rightarrow\\ \\ \Rightarrow 60^{\circ}+x=90^{\circ} \Rightarrow\boxed{x = 30^{\circ} \text{ (cand }x\text{ apartine cadranului I)}}$