\nEnunciado<\/b><\/span> Sabe-se que $sen\\,(\\alpha +\\pi )=a$.<\/p>\n
- \n
- Determine, em fun\u00e7\u00e3o de a, $sen\\,(2\\pi -\\alpha )$.<\/li>\n
- Determine os valores de $\\alpha $ $(0<\\alpha <2\\pi )$, quando $a=0,5$.<\/li>\n
- Se o \u00e2ngulo $\\alpha $ estiver compreendido entre $5\\frac{\\pi }{3}\\,rad$ e $3\\frac{\\pi }{2}\\,rad$, quem \u00e9 maior: o seu seno o o seu cosseno?<\/li>\n<\/ol>\nResolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n\nResolu\u00e7\u00e3o<\/b><\/span><\/p>\n
- \n
- Como $sen\\,(\\alpha +\\pi )=a\\Leftrightarrow sen\\,\\alpha =-a$, ent\u00e3o $sen\\,(2\\pi -\\alpha )=sen\\,(-\\alpha )=-sen\\,\\alpha =a$.
\n\u00ad<\/li>\n - Para $a=0,5$, temos \\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{matrix} \u00a0\u00a0 sen\\,\\alpha =-\\frac{1}{2} & \\wedge\u00a0 & 0<\\alpha <2\\pi\u00a0\u00a0 \\\\ \\end{matrix} & \\Leftrightarrow\u00a0 & \\begin{matrix} \u00a0\u00a0 \\alpha =\\frac{\\pi }{6}+\\pi\u00a0 & \\vee\u00a0 & \\alpha =2\\pi -\\frac{\\pi }{6}\u00a0 \\\\ \\end{matrix}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\begin{matrix} \u00a0\u00a0 \\alpha =\\frac{7\\pi }{6} & \\vee\u00a0 & \\alpha =\\frac{11\\pi }{6}\u00a0 \\\\ \\end{matrix}\u00a0 \\\\ \\end{array}\\] (Conclua as solu\u00e7\u00f5es no c\u00edrculo trigonom\u00e9trico)
![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/pag95-50.png\" "\\"CT\\"")
<\/a>\u00ad<\/li>\n- Se $\\alpha \\in \\left] 3\\frac{\\pi }{2},5\\frac{\\pi }{3} \\right[$, ent\u00e3o $sen\\,\\alpha <0$ e $\\cos \\alpha >0$.
\nLogo \u00e9 maior o seu cosseno.<\/li>\n<\/ol>\n - Se $\\alpha \\in \\left] 3\\frac{\\pi }{2},5\\frac{\\pi }{3} \\right[$, ent\u00e3o $sen\\,\\alpha <0$ e $\\cos \\alpha >0$.
- Como $sen\\,(\\alpha +\\pi )=a\\Leftrightarrow sen\\,\\alpha =-a$, ent\u00e3o $sen\\,(2\\pi -\\alpha )=sen\\,(-\\alpha )=-sen\\,\\alpha =a$.