Equações Trigonométricas
Autor: Guilherme Miguel Rosa | Publicado: 2013-01-27T21:27:00-02:00





EXERCÍCIOS RESOLVIDOS:


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1 – Encontre as soluções das equações trigonométricas seguintes:



a) 3tg x + 4√3 = 5√3 no intervalo [0, 2π]



b) cos²x – 3cos x + 2 = 0 no intervalo 0 ≤ x ≤ π



c) sen 2x – 1/2 = 0 no intervalo 0 ≤ x ≤ π





Resolução:
a) 3tg x + 4√3 = 5√3
&nbsp;&nbsp;&nbsp; 3tg x = 5√3 - 4√3
&nbsp;&nbsp;&nbsp; 3tg x = √3
&nbsp;&nbsp; &nbsp;&nbsp;tg x = √3
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3

No intervalo [0, 2π] os ângulos cuja tangente vale √3/3 são 30º e 210º.
S = {30º; 210º}





&nbsp;

b) cos²x – 3cos x + 2 = 0
&nbsp;&nbsp;&nbsp; cos x = t
&nbsp;&nbsp;&nbsp; t² - 3t + 2 = 0
&nbsp;&nbsp;&nbsp; t = 1&nbsp; e&nbsp; t = 2



Como não existe cosseno valendo 2:
cos x = 1
No intervalo 0 ≤ x ≤ π, x = 0.
S = { 0 }

&nbsp;

c) sen 2x – 1/2 = 0
&nbsp;&nbsp;&nbsp; sen 2x = 1/2

Os ângulos cujo seno vale 1/2 no intervalo 0 ≤ x ≤ π são 30º e 150º, porém o ângulo da questão é 2x, então:
2x = 30º&nbsp; e&nbsp; 2x = 150º
x = 15º&nbsp; e&nbsp; x = 75º
S = {15º; 75º} 
&nbsp;&nbsp; 

 



2 – (UFRGS) No intervalo [0, π] a equação tg x – 1 = 0:
a) não possui raízes.
b) possui uma única raiz.
c) possui apenas 2 raízes.
d) possui exatamente 4 raízes.
e) apresenta infinitas raízes. 



Resolução:
tg x – 1 = 0
tg x = 1

Os ângulos onde a tangente vale 1 são 45º e 225º, no intervalo [0, 2π], então, no intervalo [0, π] temos uma única raiz.

Gabarito Letra: B 







3 – (ITA 2012) Seja x є [0, 2 π] tal que sen(x)cos(x) = 2/5. 

Então, o produto e a soma de todos os possíveis valores 

de tg(x) são, respectivamente 

a) a) 1 e 0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

b) 1 e 5/2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
c) -1 e 0
d) 1 e 5
e) -1 e -5/2

Resolução:
sen(x)cos(x) = 2/5

Dividimos todo mundo por cos²(x), pois surge uma tangente e uma secante ao quadrado que podemos transformar em tangente depois, vejamos como fazemos aparecer o que buscamos:
sen(x)cos(x) = &nbsp;&nbsp;2/5&nbsp; 
cos(x)cos(x) = cos²(x)

tg(x) = 2 .&nbsp; 1&nbsp;&nbsp;&nbsp; .
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5&nbsp; cos²(x)

tg(x) = 2sec²(x)
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5

A identidade trigonométrica diz que:
sec²x = 1 + tg²x

tg(x) = 2/5[1 + tg²(x)]
tg(x) = 2/5 + 2/5tg²(x)
– 2/5tg²(x) + tg(x) – 2/5 = 0

Caímos numa equação do 2º grau, cuja incógnita é tg(x) fazendo a soma e o produto em função de tg(x) chegamos em:
Soma = 5/2&nbsp;&nbsp; Produto: 1

Gabarito Letra: B


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Equações Trigonométricas