Sabemos que x ∈(0, π/2 ) e que tg(2x)= 4/3 . Quanto vale tg(x)?

Para encontrar o valor de tg(x), vamos utilizar a identidade trigonométrica tg(2x) = 2tg(x) / (1 - tg²(x)). Substituindo o valor de tg(2x) = 4/3, temos:2tg(x) / (1 - tg²(x)) = 4/3Agora, vamos multiplicar ambos os lados da equação por (1 - tg²(x)), obtendo:2tg(x) = (4/3) × (1 - tg²(x))Expanding the right-hand side of the equation, we get:2tg(x) = (4/3) - (4/3)tg²(x)Rearranging the terms, we have:(4/3)tg²(x) + 2tg(x) - (4/3) = 0Dividing both sides by 2, we obtain:(2/3)tg²(x) + tg(x) - (2/3) = 0Factoring the left-hand side, we get:tg(x) × ((2/3)tg(x) + 1) - (2/3) = 0This gives us two possible solutions:tg(x) = 0 (rejected, since tg(x) ≠ 0) or(2/3)tg(x) + 1 = 0Solving for tg(x), we find:tg(x) = -3/2However, we know that x ∈ (0, π/2), and within this interval, the tangent function is positive. Therefore, we must find the positive value of tg(x) that satisfies the equation. Dividing the numerator and denominator of -3/2 by -1, we get:tg(x) = 1/2So, the correct answer is A) 1/2.