https://file.aiquickdraw.com/mathbot/file/172677310482908apvcic.jpg

The image shows a problem asking to find the values of \( x \) for which the equation:

\[
6 \cdot \tan(x) - 2 = 7
\]

is true. This is the same equation as in the previous problem you shared.

To summarize the solution:

### Steps to solve:

1. **Isolate \( \tan(x) \):**
   \[
   6 \cdot \tan(x) - 2 = 7
   \]
   Add 2 to both sides:
   \[
   6 \cdot \tan(x) = 9
   \]
   Now divide both sides by 6:
   \[
   \tan(x) = \frac{9}{6} = 1.5
   \]

2. **Find the principal solution:**
   Take the inverse tangent of 1.5:
   \[
   x = \tan^{-1}(1.5)
   \]
   Using a calculator:
   \[
   x \approx 0.9828 \, \text{radians}
   \]

3. **Find the general solution:**
   Since the tangent function is periodic with a period of \( \pi \), the general solution is given by:
   \[
   x = 0.9828 + n\pi, \quad n \in \mathbb{Z}
   \]
   where \( n \) is any integer.

### Final answer:
The values of \( x \) are \( x = 0.9828 + n\pi \), where \( n \) is any integer.

Feel free to ask for further clarifications or details. Here are 5 related questions to explore:

1. What would be the solution if the equation was \( 6 \tan(x) - 2 = -7 \)?
2. How does the period of the tangent function affect the solution of this type of equation?
3. How can you convert the solution in radians to degrees?
4. What other trigonometric functions (like sine or cosine) have similar periodicity properties?
5. Can the tangent function take any real number as an output, or are there limitations?

**Tip:** Always check the calculator mode when working with trigonometric functions. It should be in radians or degrees, depending on the question.

For which values of x is the equation 6 * tan(x) - 2 = 7 true?

Find the solution to the equation 6 * tan(x) - 2 = 7 using trigonometric principles. Understand the periodicity of the tangent function and how to apply inverse trigonometric functions. Suitable for high school students in Grades 10-12.

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