To solve the equation [tex]\(\sec(x) \cos(3x) = 0\)[/tex] within the interval [tex]\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)[/tex], let us break down the problem step by step.
### Step 1: Understand the Equation
The equation is [tex]\(\sec(x) \cos(3x) = 0\)[/tex]. Recall that the secant function is the reciprocal of the cosine function:
[tex]\[ \sec(x) = \frac{1}{\cos(x)} \][/tex]
So, our equation becomes:
[tex]\[ \frac{1}{\cos(x)} \cos(3x) = 0 \][/tex]
This implies:
[tex]\[ \cos(3x) = 0 \][/tex]
### Step 2: Solve [tex]\(\cos(3x) = 0\)[/tex]
The equation [tex]\(\cos(3x) = 0\)[/tex] is satisfied when [tex]\(3x\)[/tex] is an odd multiple of [tex]\(\frac{\pi}{2}\)[/tex]:
[tex]\[ 3x = \left(2k + 1\right) \frac{\pi}{2} \][/tex]
where [tex]\(k\)[/tex] is an integer.
### Step 3: Solve for [tex]\(x\)[/tex]
To find [tex]\(x\)[/tex], solve the equation for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\left(2k + 1\right)\pi}{6} \][/tex]
### Step 4: Determine [tex]\(x\)[/tex] within the given interval [tex]\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)[/tex]
We need to find values of [tex]\(k\)[/tex] such that [tex]\(\frac{\left(2k + 1\right)\pi}{6}\)[/tex] lies within [tex]\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)[/tex].
Solving the inequalities:
[tex]\[ -\frac{\pi}{2} \leq \frac{\left(2k + 1\right)\pi}{6} \leq \frac{\pi}{2} \][/tex]
### For [tex]\(k = -1\)[/tex]:
[tex]\[ x = \frac{\left(2(-1) + 1\right)\pi}{6} = \frac{-\pi}{6} \][/tex]
### For [tex]\(k = 0\)[/tex]:
[tex]\[ x = \frac{\left(2(0) + 1\right)\pi}{6} = \frac{\pi}{6} \][/tex]
These values fall within the interval [tex]\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)[/tex].
### Step 5: Verify Solutions
Ensure [tex]\(\cos(x) \neq 0\)[/tex] at these solutions to prevent division by zero in [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex]:
- For [tex]\(x = -\frac{\pi}{6}\)[/tex], [tex]\(\cos(-\frac{\pi}{6}) = \cos(\frac{\pi}{6}) \neq 0\)[/tex].
- For [tex]\(x = \frac{\pi}{6}\)[/tex], [tex]\(\cos(\frac{\pi}{6}) \neq 0\)[/tex].
Thus, the solutions to the equation [tex]\(\sec(x) \cos(3x) = 0\)[/tex] in the interval [tex]\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)[/tex] are:
[tex]\[ \boxed{-\frac{\pi}{6}, \frac{\pi}{6}} \][/tex]
