Question

sinx=3/4 si x apartine(pi/2;pi) calculati sinx+cosx

Answer 1

$x\in \Big(\dfrac{\pi}{2},\pi\Big)\\ \\ \sin x = \dfrac{3}{4}\\ \\ \sin^2 x+\cos^2 x = 1 \Rightarrow \cos^2 x = 1-\sin^2 x \Rightarrow \cos^2 x = 1-\Big(\dfrac{3}{4}\Big)^2 \Rightarrow \\ \\ \Rightarrow \cos^2 x = 1-\dfrac{9}{16} \Rightarrow \xos^2 x = \dfrac{16-9}{16} \Rightarrow \cos^2 x = \dfrac{7}{16} \Rightarrow \\ \\ \Rightarrow \cos x = \pm\sqrt{\dfrac{7}{16}} \Rightarrow \cos x = \pm \dfrac{\sqrt7}{4}, dar , x\in \Big(\dfrac{\pi}{2},\pi\Big) \Rightarrow$

$\\ \\ \Rightarrow \cos x = -\dfrac{\sqrt7}{4}\\ \\ \\ \sin x+\cos x = \dfrac{3}{4}-\dfrac{\sqrt7}{4} \Rightarrow \boxed{\sin x +\cos x = \dfrac{3-\sqrt7}{4}}$