Let's solve the given equation step by step:
Given:
[tex]\[ 3 \sec x - 2 = 1 \][/tex]
1. Add 2 to both sides of the equation:
[tex]\[ 3 \sec x = 3 \][/tex]
2. Divide both sides by 3:
[tex]\[ \sec x = 1 \][/tex]
3. Recall that the secant function is the reciprocal of the cosine function:
[tex]\[ \sec x = \frac{1}{\cos x} \][/tex]
So, if [tex]\(\sec x = 1\)[/tex], then:
[tex]\[ \frac{1}{\cos x} = 1 \][/tex]
This implies:
[tex]\[ \cos x = 1 \][/tex]
4. Determine the values of [tex]\(x\)[/tex] for which [tex]\(\cos x = 1\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex]. The cosine function equals 1 at:
[tex]\[ x = 0 \][/tex]
So, the solution to the equation [tex]\(3 \sec x - 2 = 1\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[ x = 0 \][/tex]
Therefore, the correct choice is:
C. 0
