Oblicz wartość wyrażenia-Trygonometria

[Maniek654]

Oblicz wartość wyrażenia \(\sin (x-y)\),mając dane \(\sin x=\frac{3}{4},x\in (90^{\circ},180^{\circ})\) oraz \(y=-1230^{\circ}\)

[eresh]

Oblicz wartość wyrażenia \(\sin (x-y)\),mając dane \(\sin x=\frac{3}{4},x\in (90^{\circ},180^{\circ})\) oraz \(y=-1230^{\circ}\)

\(\sin (x-y)=\sin x\cos y-\cos x\sin y\\
\sin x=\frac{3}{4}\\
\cos x=-\sqrt{1-(\frac{3}{4})^2}\\
\cos x=\frac{-\sqrt{7}}{4}\\
\sin y=\sin (-1230^{\circ})=-\sin (3\cdot 360^{\circ}+150^{\circ})=-\sin 150^{\circ}=-\sin 30^{\circ}=-0,5\\
\cos y=\cos(-1230^{\circ})=\cos (3\cdot 360^{\circ}+150^{\circ})=\cos 150^{\circ}=-\cos 30^{\circ}=-\frac{\sqrt{3}}{2}\\
\sin (x-y)=\frac{3}{4}\cdot (-\frac{\sqrt{3}}{2})+\frac{\sqrt{7}}{4}\cdot (-\frac{1}{2})=\frac{-3\sqrt{3}-\sqrt{7}}{8}\)