{"id":690,"date":"2020-03-29T18:10:05","date_gmt":"2020-03-29T21:10:05","guid":{"rendered":"http:\/\/cinoto.com.br\/matematica\/?p=690"},"modified":"2026-01-20T21:46:38","modified_gmt":"2026-01-21T00:46:38","slug":"demonstre-que-se-a-b-c-sao-angulos-internos-de-um-triangulo-nao-retangulo","status":"publish","type":"post","link":"http:\/\/cinoto.com.br\/matematica\/demonstre-que-se-a-b-c-sao-angulos-internos-de-um-triangulo-nao-retangulo\/","title":{"rendered":"2) Demonstre que, se A, B, C s\u00e3o \u00e2ngulos internos de um tri\u00e2ngulo n\u00e3o ret\u00e2ngulo, vale a rela\u00e7\u00e3o: 1\/(tg A).(tg B) + 1\/(tg B).(tg C) + 1\/(tg C).(tg A) = 1"},"content":{"rendered":"\n

com A, B, C diferente de Pi\/2.<\/p>\n\n\n\n\n\n\n\n

Resolu\u00e7\u00e3o:<\/h2>\n\n\n\n

Vamos usar a f\u00f3rmula da tangente da diferen\u00e7a de arcos para deduzir o seguinte:
A + B + C = 180\u00b0
A = 180\u00b0 – (B + C)
tg A = tg[180\u00b0 – (B + C)]
tg A = [tg 180\u00b0 – tg(B + C)]\/[1 + (tg 180\u00b0) . tg (B + C)]
tg A = [0 – tg(B + C)]\/[1 + 0 . tg (B + C)]
tg A = -tg(B + C)\/1
tg A = -tg(B + C)<\/p>\n\n\n\n

E sabemos que:
tg(B + C) = (tg B + tg C)\/[(1 – (tg B).(tg C)]
-tg(B + C) = (tg B + tg C)\/[(tg B).(tg C) – 1]<\/p>\n\n\n\n

Ent\u00e3o colocando isso no primeiro membro da equa\u00e7\u00e3o que temos:
= primeiro membro
= 1\/(tg A).(tg B) + 1\/(tg B).(tg C) + 1\/(tg C).(tg A)
= 1\/{[-tg (B + C)].(tg B)} + 1\/(tg B).(tg C) + 1\/{(tg C).[-tg (B + C)]}
= 1\/{(tg B + tg C)\/[(tg B).(tg C) – 1]}.(tg B) + 1\/(tg B).(tg C) +
+ 1\/(tg C).{(tg B + tg C)\/[(tg B).(tg C) – 1]}
= 1\/{[tg2<\/sup> B + (tg C).(tg B)]\/[(tg B).(tg C) – 1]} + 1\/(tg B).(tg C) +
+ 1\/{[(tg C).(tg B) + tg2<\/sup> C]\/[(tg B).(tg C) – 1]}
= [(tg B).(tg C) – 1]\/[tg2<\/sup> B + (tg C).(tg B)] + 1\/(tg B).(tg C) +
+ [(tg B).(tg C) – 1]\/[(tg C).(tg B) + tg2<\/sup> C]
= (tg C).[(tg B).(tg C) – 1]\/[(tg2<\/sup> B).(tg C) + (tg2<\/sup> C).(tg B)] + (tg B + tg C)\/[(tg2<\/sup> B).(tg C) +
+ (tg2<\/sup> C).(tg B)] + (tg B).[(tg B).(tg C) – 1]\/[(tg2<\/sup> B).(tg C) + (tg2<\/sup> C).(tg B)]
= [(tg2<\/sup> C).(tg B) – tg C + tg B + tg C + (tg2<\/sup> B).(tg C) – tg B]\/[(tg2<\/sup> B).(tg C) + (tg2<\/sup> C).(tg B)]
= [(tg2<\/sup> C).(tg B) + (tg2<\/sup> B).(tg C)]\/[(tg2<\/sup> B).(tg C) + (tg2<\/sup> C).(tg B)]
= 1
= 2\u00ba membro<\/p>\n","protected":false},"excerpt":{"rendered":"

com A, B, C diferente de Pi\/2.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[26],"tags":[30],"class_list":["post-690","post","type-post","status-publish","format-standard","hentry","category-trigonometria","tag-insano"],"_links":{"self":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/690","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/comments?post=690"}],"version-history":[{"count":1,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/690\/revisions"}],"predecessor-version":[{"id":691,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/posts\/690\/revisions\/691"}],"wp:attachment":[{"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/media?parent=690"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/categories?post=690"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/cinoto.com.br\/matematica\/wp-json\/wp\/v2\/tags?post=690"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}