Question

Se x é a medida do arco em radianos e α é um número real, determine α sabendo que sen x = √3 - α e cos x = α -2/2
( COLOQUEM OS CÁLCULOS POR FAVOR )

Answer 1

$\displaystyle \underline{\text{Rela{\c c}{\~a}o fundamental da trigonometria}}: \\\\ \text{sen}^2(\text x)+\text{cos}^2(\text x) = 1 \\\\\ \text{temos} :\\\\ 1) \ \text{sen(x)}=\sqrt{3-\text a} \\\\ \underline{\text{condi{\c c}{\~a}o de exist{\^e}ncia do seno/restri{\c c}{\~a}o para a }}:\\\\ 3-\text a \geq 0 \to \boxed{\text a \leq 3} \\\\\\ 2) \ \text{cos(x)}=\frac{\text a-2}{2}$

Substituindo na relação fundamental da trigonometria :

$\displaystyle (\sqrt{3-\text a})^2+(\frac{\text a-2}{2})^2=1 \\\\\\ 3-\text a+\frac{\text a^2-4\text a+4}{4}=1 \\\\\\ 12 -4\text a+\text a^2-4\text a+4=4 \\\\ \text a^2-8\text a+16=4 \\\\ (\text a-4)^2=4 \\\\ \text a-4 = \pm 2 \\\\ \text a = 4+2 \to \text a = 6 \to \text{(N{\~a}o pode pq foge da restri{\c c}{\~a}o)} \\\\ \text a =4-2 \to \text a = 2 \checkmark \\\\ \text{Portanto} \\\\ \huge\boxed{\ \text a =2\ }\checkmark$