Question

Calcule todos os possíveis valores para x que satisfaçam a:

a) sen x = 1

b) 2cos² x + cos x = 1

c) sen x - sen³ x = 0​

Answer 1

Resposta:

Explicação passo-a-passo:

k∈Z

a)

$senx=1\\\\x=arcsen1\\\\x=\dfrac{\pi}{2}+2k\pi$

b)

$2cos^{2}x+cosx=1\\\\2cos^{2}x+cosx-1=\\\\cosx=\dfrac{-1\pm\sqrt{1^{2}-4.2.(-1)}}{2.2} \\\\cosx=\dfrac{-1\pm\sqrt{1+8}}{4}\\\\cosx=\dfrac{-1\pm\sqrt{9}}{4}\\\\cosx=\dfrac{-1\pm3}{4}\\\\cosx=\dfrac{-1+3}{4}=\dfrac{2}{4}=\dfrac{1}{2}\\\\x=arccos\frac{1}{2}\\\\x=\dfrac{\pi}{3}+2k\pi\\\\ou\\\\cosx=\dfrac{-1-3}{4}=\dfrac{-4}{4}=-1\\\\x=arccos(-1)\\\\x=\pi+2k\pi$

c)

$senx-sen^{3}x=0\\ \\senx.(1-sen^{2}x)=0\\ \\senx(1-senx).(1+senx)=0\\\\senx=0\\\\x=arcsen0\\\\x=0+2k\pi\\\\ou\\\\1-senx=0\\\\senx=1\\\\x=arcsen1\\\\x=\dfrac{\pi}{2}+2k\pi\\ \\ou\\\\1+senx=0\\\\senx=-1\\\\x=arcsen(-1)\\\\x=\dfrac{3\pi}{2}+2k\pi$