Resolva as seguintes equa\u00e7\u00f5es trigonom\u00e9tricas, no intervalo indicado:<\/p>\n
1.<\/strong> 2.<\/strong> 3.<\/strong> 4.<\/strong>
\nOra,
\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 -\\sqrt{3}-2\\,sen\\,\\theta =0 & \\Leftrightarrow\u00a0 & sen\\,\\theta =-\\frac{\\sqrt{3}}{2}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}} \u00a0\u00a0 \\theta =-\\frac{\\pi }{3}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -(-\\frac{\\pi }{3}))+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{array}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}} \u00a0\u00a0 \\theta =-\\frac{\\pi }{3}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{4\\pi }{3}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{array}\u00a0 \\\\ \\end{array}\\] donde, para os valores indicados de k, se obt\u00e9m: \\[\\begin{matrix} \u00a0\u00a0 k=-1: & \\theta =-\\frac{7\\pi }{3} & \\vee\u00a0 & \\theta =-\\frac{2\\pi }{3}\u00a0 \\\\ \u00a0\u00a0 k=0: & \\theta =-\\frac{\\pi }{3} & \\vee\u00a0 & \\theta =\\frac{4\\pi }{3}\u00a0 \\\\ \u00a0\u00a0 k=1: & \\theta =\\frac{5\\pi }{3} & \\vee\u00a0 & \\theta =\\frac{10\\pi }{3}\u00a0 \\\\ \\end{matrix}\\] Logo, no intervalo considerado, o conjunto-solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9 $S=\\left\\{ -\\frac{2\\pi }{3},-\\frac{\\pi }{3} \\right\\}$.
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\nOra,
\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 -2+\\sqrt{3}\\,tg\\,\\theta =1 & \\Leftrightarrow\u00a0 & tg\\,\\theta =\\frac{3}{\\sqrt{3}}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & tg\\,\\theta =\\sqrt{3}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\theta =\\frac{\\pi }{3}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{array}\\] donde, para os valores indicados de k, se obt\u00e9m: \\[\\begin{matrix} \u00a0\u00a0 k=0: & \\theta =\\frac{\\pi }{3}\u00a0 \\\\ \u00a0\u00a0 k=1: & \\theta =\\frac{4\\pi }{3}\u00a0 \\\\ \u00a0\u00a0 k=2: & \\theta =\\frac{7\\pi }{3}\u00a0 \\\\ \\end{matrix}\\] Logo, no intervalo considerado, o conjunto-solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9 $S=\\left\\{ \\frac{\\pi }{3},\\frac{4\\pi }{3} \\right\\}$.
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 1+\\sqrt{2}\\cos \\theta =3 & \\Leftrightarrow\u00a0 & \\cos \\theta =\\frac{2}{\\sqrt{2}}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\cos \\theta =\\sqrt{2}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\theta \\in \\left\\{ {} \\right\\}\u00a0 \\\\ \\end{array}\\] Logo, a equa\u00e7\u00e3o \u00e9 imposs\u00edvel.
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 4{{\\cos }^{2}}\\theta =3 & \\Leftrightarrow\u00a0 & {{\\cos }^{2}}\\theta =\\frac{3}{4}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}} \u00a0\u00a0 \\cos \\theta =-\\frac{\\sqrt{3}}{2} & \\vee\u00a0 & \\cos \\theta =\\frac{\\sqrt{3}}{2}\u00a0 \\\\ \\end{array}\u00a0 \\\\ \u00a0\u00a0 {} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}} \u00a0\u00a0 \\theta =\\mp \\frac{5\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\mp \\frac{\\pi }{6}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\ \\end{array}\u00a0 \\\\ \\end{array}\\] donde, para os valores indicados de k, se obt\u00e9m: \\[\\begin{matrix} \u00a0\u00a0 k=-1: & \\theta =-\\frac{17\\pi }{6} & \\vee\u00a0 & \\theta =-\\frac{7\\pi }{6} & \\vee\u00a0 & \\theta =-\\frac{13\\pi }{6} & \\vee\u00a0 & \\theta =-\\frac{11\\pi }{6}\u00a0 \\\\ \u00a0\u00a0 k=0: & \\theta =-\\frac{5\\pi }{6} & \\vee\u00a0 & \\theta =\\frac{5\\pi }{6} & \\vee\u00a0 & \\theta =-\\frac{\\pi }{6} & \\vee\u00a0 & \\theta =\\frac{\\pi }{6}\u00a0 \\\\ \u00a0\u00a0 k=1: & \\theta =\\frac{7\\pi }{6} & \\vee\u00a0 & \\theta =\\frac{17\\pi }{6} & \\vee\u00a0 & \\theta =\\frac{11\\pi }{6} & \\vee\u00a0 & \\theta =\\frac{13\\pi }{6}\u00a0 \\\\ \\end{matrix}\\] Logo, no intervalo considerado, o conjunto-solu\u00e7\u00e3o da equa\u00e7\u00e3o \u00e9 $S=\\left\\{ -\\frac{5\\pi }{6},-\\frac{\\pi }{6},\\frac{\\pi }{6},\\frac{5\\pi }{6} \\right\\}$.<\/p>\n