\(\text{A = 3 s i n α + 4 c o s α }\)
\(\rightarrow A^2=\left(3sina+4cos\right)^2\le\left(3^2+4^2\right)\left(sin^2+cos^2a\right)=5^2\)
\(\rightarrow-5\le A\le5\)
\(\rightarrow Max_A=5\leftrightarrow\frac{sin\alpha}{3}=\frac{cos\alpha}{4}\left(\alpha< 90^o\right)\)
\(\rightarrow tan\alpha=\frac{sin\alpha}{cos\alpha}=\frac{3}{4}\rightarrow\text{a r c t a n }\frac{3}{4}\)