To solve the equation [tex]\((\sec x - \sqrt{2})(\sin x - 4) = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], we need to consider the solutions of each factor set to zero individually.
#### Step 1: Solve [tex]\(\sec x - \sqrt{2} = 0\)[/tex]
[tex]\[
\sec x - \sqrt{2} = 0 \implies \sec x = \sqrt{2}
\][/tex]
Recall that [tex]\(\sec x = \frac{1}{\cos x}\)[/tex], so we have:
[tex]\[
\frac{1}{\cos x} = \sqrt{2} \implies \cos x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
The cosine function [tex]\(\cos x = \frac{\sqrt{2}}{2}\)[/tex] has solutions at:
[tex]\[
x = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad x = -\frac{\pi}{4} + 2k\pi \quad \text{for} \quad k \in \mathbb{Z}
\][/tex]
To find solutions within the interval [tex]\([0, 2\pi)\)[/tex], we consider:
[tex]\[
x = \frac{\pi}{4}, \quad x = \frac{7\pi}{4}
\][/tex]
However, since [tex]\(\frac{7\pi}{4} - 2\pi\)[/tex] results in an angle outside the interval [tex]\([0, 2\pi)\)[/tex], the valid solution for the interval is:
[tex]\[
x = \frac{\pi}{4}
\][/tex]
#### Step 2: Solve [tex]\(\sin x - 4 = 0\)[/tex]
[tex]\[
\sin x - 4 = 0 \implies \sin x = 4
\][/tex]
The sine function, [tex]\(\sin x\)[/tex], ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. Since [tex]\(4\)[/tex] lies outside this range, there are no valid solutions for this equation.
#### Step 3: Combine the solutions
The only valid solution for the original equation in the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[
x = \frac{\pi}{4}
\][/tex]
Therefore, the solution is:
[tex]\[
x = \frac{\pi}{4}
\][/tex]
