Question

Sendo a função g(x)=1/2+1/2sec(x-pi/2) resolva a equação g(x)=3/2 para x pertencendo ao domínio(g) interseção[-pi,3pi]

Answer 1

$g(x)=\dfrac{1}{2}+\dfrac{1}{2}\sec\left(x-\dfrac{\pi}{2}\right)\\\\\\\dfrac{3}{2}=\dfrac{1}{2}+\dfrac{1}{2}\sec\left(x-\dfrac{\pi}{2}\right)\\\\\\\\\dfrac{3}{2}-\dfrac{1}{2}=\dfrac{1}{2}\sec\left(x-\dfrac{\pi}{2}\right)\\\\\\1=\dfrac{1}{2}\sec\left(x-\dfrac{\pi}{2}\right)\\\\\\2=\sec\left(x-\dfrac{\pi}{2}\right)\\\\\\2=\dfrac{1}{\cos\left(x-\dfrac{\pi}{2}\right)}\\\\\\2=\dfrac{1}{\cos(x)\cdot\cos\left(\dfrac{\pi}{2}\right)+\sin(x)\cdot\sin\left(\dfrac{\pi}{2}\right)}$

$2=\dfrac{1}{\cos(x)\cdot\cos\left(\dfrac{\pi}{2}\right)+\sin(x)\cdot\sin\left(\dfrac{\pi}{2}\right)}\\\\\\2=\dfrac{1}{\sin(x)}\\\\\\\sin(x)=\dfrac{1}{2}\\\\\\\sin\left(\dfrac{\pi}{6}\right)=\dfrac{1}{2}\\\\\\\\\\g\left(\dfrac{\pi}{6}\right)=\dfrac{3}{2}\\\\\\\\\\\boxed{x=\dfrac{\pi}{6}}$