$$ 1 \leftarrow \quad $$ Solve the equation for $$ x $$, where $$ x $$ is restricted to the given interval. $$ y=8 \sec 3 x \text {, for } x \text { in }\left[0, \frac{\pi}{6}\right) \cup\left(\frac{\pi}{6}, \frac{\pi}{3}\right] $$

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Solve the equation $$ 8 \sec(3x)=1 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$8\sec\left(3xight)=1$$
- step1: Find the domain:
  $$8\sec\left(3xight)=1,x
eq \frac{\pi }{6}+\frac{k\pi }{3},k \in \mathbb{Z}$$
- step2: Multiply both sides:
  $$8\sec\left(3xight)	imes \frac{1}{8}=1	imes \frac{1}{8}$$
- step3: Calculate:
  $$\sec\left(3xight)=1	imes \frac{1}{8}$$
- step4: Calculate:
  $$\sec\left(3xight)=\frac{1}{8}$$
- step5: Use the inverse trigonometric function:
  $$3x=\operatorname{arcsec}\left(\frac{1}{8}ight)$$
- step6: Calculate:
  $$x 
otin \mathbb{R}$$
- step7: Check if the solution is in the defined range:
  $$x 
otin \mathbb{R},x
eq \frac{\pi }{6}+\frac{k\pi }{3},k \in \mathbb{Z}$$
- step8: Find the intersection:
  $$x 
otin \mathbb{R}$$
The equation $$8 \sec(3x) = 1$$ has no real solution for $$x$$ in the given interval $$[0, \frac{\pi}{6}) \cup (\frac{\pi}{6}, \frac{\pi}{3}]$$.
The equation $$8 \sec(3x)=1$$ has no real solution in the given interval.

Solve the equation for , where is restricted to the given interval.

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