Soma das soluções trigonométricas.

por Betoneira de Natal Qui 17 Mar 2022, 00:02

Encontre a soma das raízes de:

Soma das soluções trigonométricas. Svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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Resp.: Sem gabarito.

Não consigo achar as raízes, só consegui 1 delas. Segue oq eu fiz:

Aí então:
sen(2x) = 0 ------> x = kπ/2 , k ∈ ℤ
Mas eu só acho essa, não consigo relacionar o: cos(x) + cos(7x) = 0

por João Pedro Lima Qui 17 Mar 2022, 00:26

Fala, Betoneira!
Já está de noite por agora, vou dar a ideia. Se não sair me chama amanhã.
O problema é resolver:
cos(x) + cos(7x) = 0, certo?

Usa a seguinte fórmula:
[latex]cos(\alpha)+cos(\beta) = 2cos(\frac{\alpha+\beta}{2})cos(\frac{\alpha-\beta}{2})[/latex]

Se tiver problemas para decorar a fórmula, abre cos(x+y) + cos(x-y) e depois chama x+y de alpha e x-y de beta.

por **Elcioschin** Qui 17 Mar 2022, 10:24

Este método é denominadp Prostaférese (transformação de soma em produto)

Uma sugestão é inverter os termos, para facilitar as contas:

cos(7x) + cosx = 2.cos[(7.x + x)/2].cos[(7.x - x)/2] = 2.cos(4.x).cos(3.x) = 0

por Betoneira de Natal Qui 17 Mar 2022, 22:03

Olá
Perdão pela demora, tive um problema na internet aqui onde moro, então n pude responder hj de manhã.

Eu havia me esquecido por um momento que eu podia aplicar Prostaférese(não aprendi c/ esse nome, interessante.)

Então, resolvendo o restinho:
cos.(4x) = 0 --> 4x = π/2 + kπ , k ∈ ℤ ---> x = π/8 + kπ , k ∈ ℤ
cos.(3x) = 0 --> 3x = π/2 + kπ , k ∈ ℤ ---> x = π/6 + kr , k ∈ ℤ

Creio que seja isso.
Obrigado!