Answer:
[tex]\theta=\dfrac{\pi }{3}\\\\\theta=\dfrac{5\pi }{3}[/tex]
Step-by-step explanation:
[tex]\theta = x[/tex]
[tex]5-4 \sec (\theta) = -3[/tex]
[tex]-4 \sec (\theta) = -3-5[/tex]
[tex]-4 \sec (\theta) = -8[/tex]
[tex]\dfrac{-4 \sec (\theta)}{-4} =\dfrac{-8}{-4}[/tex]
[tex]\sec (\theta)=2[/tex]
Once
[tex]\sec(\theta)=\dfrac{1}{\cos(\theta)}[/tex]
[tex]\sec (\theta)=2 \implies \dfrac{1}{\cos(\theta)}=2 \implies \cos(\theta)=\dfrac{1}{2}[/tex]
The solutions for [tex]\cos(\theta)=\dfrac{1}{2}[/tex] are
[tex]\theta=\dfrac{\pi }{3}+2\pi n, n \in \mathbb{Z}\\\\\theta=\dfrac{5\pi }{3}+2\pi n, n \in \mathbb{Z}[/tex]
But once the interval for the solution was given,
For [tex][0, 2\pi)[/tex]
[tex]\theta=\dfrac{\pi }{3}\\\\\theta=\dfrac{5\pi }{3}[/tex]
