Question

valores de m para q possa ter ​

Answer 1

Tanto a função Seno quanto a função Cosseno estão limitadas no intervalo de [ -1 , 1 ]. Sendo assim vamos analisar o problema :

1)

$\displaystyle \text {Sen(x)} = \frac{3+ \text m }{4}$

Analisando o intervalo da função seno :

$\displaystyle -1\leq \frac{3+\text m }{4} \leq 1$

Direita :

$\displaystyle \frac{3+\text m}{4} \leq 1 \\\\ 3+ \text m \leq 4 \\\\\boxed{\text m \leq 1 }$

Esquerda :

$\displaystyle \frac{3+\text m}{4} \geq -1 \\\\ 3+ \text m \geq -4 \\\\\boxed{\text m \geq -7 }$

Então :

$\huge\boxed{-7 \leq \text m \leq 1 }\checkmark$

o intervalo já representa todos os valores reais de m.

 

2)

$\displaystyle \text{cos(x)} = \frac{2\text m + 7}{6}$

analisando o intervalo da função cosseno :

$\displaystyle -1\leq \frac{2\text m + 7}{6} \leq 1$

Direita :

$\displaystyle \frac{2\text m + 7}{6} \leq 1 \\\\ 2\text m + 7 \leq 6 \\\\ 2\text m \leq -1 \\\\ \boxed{\text m \leq \frac{-1}{2}}$

Esquerda :

$\displaystyle \frac{2\text m + 7}{6} \geq -1 \\\\ 2\text m + 7 \geq -6 \\\\ 2\text m \geq -13 \\\\ \boxed{\text m \geq \frac{-13}{2}}$

Portanto :

$\displaystyle \huge\boxed{\frac{-13}{2}\leq \text m \leq \frac{-1}{2}}\checkmark$