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Exercício de Relações entre as razões trigonométricas
Exercício:
Determine a relação entre os números reais “a” e “b”, de modo que as igualdades 1+cos x= a.sen x e 1-cos x= b.sen x, com x na primeira volta, sejam satisfeitas simultaneamente.
1) Isolando cos x de cada equação:
> 1 + cos x = a.sen x => cos x = a.sen x – 1 (I)
> 1 – cos x = b.sen x => cos x = 1 – b.senx (II)
2) cos x = cox => a.senx – 1 = 1 – b.sen x => a.sen x + b.sen x = 2 => sen x(a + b) = 2 => sen x = 2/(a + b)
3) Como x pertence à primeira volta no ciclo trigonométrica:
0 ≤ sen x ≤ 1 => 0 ≤ 2/(a + b) ≤ 1 .(a + b) => 0 < 2 ≤ (a + b) => a+b ≥ 2
Obs: Avise-me caso seja outra resposta.
(fmu/fiam/faam-sp) se tg a = 2 e a é um arco do 3 quadrante, sen a vale:
tg 5π/4 – cotg5π/2
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