Esercizio 30 
. Risolvere i seguenti sistemi:
![\[(1)\quad\left\{\begin{array}{l} \sin x+2\sqrt{2}\cos y=5/2\\ 3\sin x-\sqrt{2}\cos y=1/2 \end{array}\right.\qquad (2)\quad\left\{\begin{array}{l} x-y=\pi/2\\ 4\cos x\cos y+\sqrt{3}=0 \end{array}\right..\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-10103866e43cdd4f987cf2159597c8dd_l3.png "Rendered by QuickLaTeX.com")
Svolgimento.
Sommando alla prima la seconda equazione moltiplicata per 
otteniamo
![\[7\sin x=\frac{7}{2}\quad \Leftrightarrow \quad \sin x=\frac{1}{2},\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-fc551df5649f80c0514f7ddae42ff9b7_l3.png "Rendered by QuickLaTeX.com")
le cui soluzioni sono
![\[x=\frac{\pi}{6}+2k\pi,\qquad x=\frac{5\pi}{6}+2k\pi.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-c578f9b74930031f3d11f337474c1bc9_l3.png "Rendered by QuickLaTeX.com")
Sostituendo tale valore di 
nella prima si ha
![\[\cos x=\frac{\sqrt{2}}{2},\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-6e49d69d89bbbaf29e070ce534231cdb_l3.png "Rendered by QuickLaTeX.com")
le cui soluzioni sono
![\[y=\pm\frac{\pi}{4}+2k\pi.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-cde909ff0836a9b3ba497bd063ac5cec_l3.png "Rendered by QuickLaTeX.com")
Poiché 
, si ha per la seconda equazione
![\[-4\sin y\cos y+\sqrt{3}=0,\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-bcfd9ec0eaabf025bb804564da119349_l3.png "Rendered by QuickLaTeX.com")
che, mediante la formula di duplicazione del seno diviene
![\[\sin 2y=\frac{\sqrt{3}}{2},\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-d7009ca57e334afffd9fe9d4da8dbe5b_l3.png "Rendered by QuickLaTeX.com")
le cui soluzioni sono
![\[2y=\frac{\pi}{3}+2k\pi,\qquad 2y=\frac{2\pi}{3}+2k\pi,\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-4785fa9c6ebe1a372d4f2ddfd74abc66_l3.png "Rendered by QuickLaTeX.com")
e quindi
![\[y=\frac{\pi}{6}+k\pi\quad \Leftrightarrow \quad x=\frac{2\pi}{3}+k\pi,\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-997068378157ac0bc9bec04fe164a008_l3.png "Rendered by QuickLaTeX.com")
![\[y=\frac{\pi}{3}+k\pi\quad \Leftrightarrow \quad x=\frac{5\pi}{6}+k\pi.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-a9229e70b1ad56c05cff1ba9fcfcdbfd_l3.png "Rendered by QuickLaTeX.com")
Fonte: ignota.