{"id":216686,"date":"2024-05-05T04:28:51","date_gmt":"2024-05-05T07:28:51","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=216686"},"modified":"2024-05-07T02:32:39","modified_gmt":"2024-05-07T05:32:39","slug":"problemao-uma-soma-fascinante","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-uma-soma-fascinante\/","title":{"rendered":".Desafio: Uma soma fascinante!"},"content":{"rendered":"

✏ Link do problema para dispositivos da Apple<\/em>.<\/span><\/a><\/strong><\/p>\n

\n

Problema
\n (Indicado a partir do 2\u00ba ano do E. M.)<\/span><\/p>\n

\nCalcule o valor da soma
\n[tex]\\qquad 2 \\ \\mathrm{sen}\\, 2^\\circ+ 4 \\ \\mathrm{sen} \\,4^\\circ+ 6 \\ \\mathrm{sen}\\, 6^\\circ+\\cdots +178 \\ \\mathrm{sen} \\,178^\\circ[\/tex].
\nSe necess\u00e1rio, use [tex] \\mathrm{cotg} \\ 1^\\circ=57,29[\/tex].<\/p>\n

Extra\u00eddo de XXVI OPM – N\u00edvel Gama<\/strong>.<\/p>\n<\/div>\n

 <\/p>\n

\n\"explicador_p\"<\/p>\n

Lembretes<\/p>\n

(I)<\/strong> <\/span> Dados dois \u00e2ngulos [tex]a[\/tex] e [tex]b[\/tex], com [tex]0^{\\circ}\\lt a\\lt 180^{\\circ}[\/tex] e [tex]0^{\\circ}\\lt b\\lt 180^{\\circ}[\/tex], vale a rela\u00e7\u00e3o [tex]2 \\ \\mathrm{sen} \\, a \\ \\mathrm{sen} \\, b=\\cos (a-b)-\\cos(a+b)[\/tex].
\n (II)<\/strong> <\/span> Dado um \u00e2ngulo [tex]a[\/tex], com [tex]0^{\\circ}\\lt a\\lt 180^{\\circ}[\/tex], ent\u00e3o [tex]\\cos(180^\\circ-a)=-\\cos a[\/tex].\n<\/div>\n

 <\/p>\n

\n

Solu\u00e7\u00e3o<\/p>\n

\nVamos denotar a soma a ser calculada por [tex]S[\/tex].
\nAssim, multiplicando [tex]S[\/tex] por [tex]\\mathrm{sen} \\, 1^\\circ[\/tex], temos
\n[tex]\\quad S \\cdot \\mathrm{sen} \\,1^\\circ=2 \\ \\mathrm{sen} \\, 2^\\circ \\cdot \\mathrm{sen} \\, 1^\\circ+ 4 \\ \\mathrm{sen} \\, 4^\\circ \\cdot \\mathrm{sen} \\, 1^\\circ+ 6 \\ \\mathrm{sen} \\, 6^\\circ \\cdot \\mathrm{sen} \\, 1^\\circ+\\cdots \\\\
\n\\qquad \\qquad \\cdots+178 \\ \\mathrm{sen} \\, 178^\\circ \\cdot \\mathrm{sen} \\, 1^\\circ.[\/tex]<\/p>\n

\n

Utilizando o fato de que [tex]2 \\ \\mathrm{sen} \\, a \\ \\cdot \\mathrm{sen} \\, b=\\cos (a-b)-\\cos(a+b)[\/tex], obtemos
\n[tex]\\quad S \\cdot \\mathrm{sen} \\,1^\\circ=\\cos1^\\circ-\\cos3^\\circ+2(\\cos3^\\circ-\\cos5^\\circ)+3(\\cos5^\\circ-\\cos7^\\circ)+\\dots\\\\
\n\\qquad \\qquad \\cdots+89(\\cos177^\\circ-\\cos179^\\circ)[\/tex]
\n[tex]\\quad S \\cdot \\mathrm{sen} \\,1^\\circ=\\cos1^\\circ-\\cos3^\\circ+2 \\cos3^\\circ-2 \\cos5^\\circ+3 \\cos5^\\circ-3 \\cos7^\\circ+\\dots\\\\
\n\\qquad \\qquad \\cdots+89 \\cos177^\\circ-89 \\cos179^\\circ[\/tex]
\n[tex]\\quad S \\cdot \\mathrm{sen} \\,1^\\circ=\\cos1^\\circ+\\cos3^\\circ+\\cos5^\\circ+\\cos7^\\circ+\\dots\\\\
\n\\qquad \\qquad \\cdots+\\cos177^\\circ-89\\cos179^\\circ.[\/tex]<\/p>\n

Com a identidade trigonom\u00e9trica [tex]~\\cos(180^\\circ-a)=-\\cos a~[\/tex] podemos obter<\/p>\n