Let's break down the equation step by step.

You have the equation:

\[
6\sin(x) = 2(3)
\]

1. Simplify the right-hand side:
   \[
   2(3) = 6
   \]
   So, the equation becomes:
   \[
   6\sin(x) = 6
   \]

2. Now, divide both sides of the equation by 6 to isolate \(\sin(x)\):
   \[
   \sin(x) = \frac{6}{6} = 1
   \]

3. The equation is now \(\sin(x) = 1\). The sine function equals 1 at specific values of \(x\), namely:
   \[
   x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}
   \]
   where \(k\) is any integer, representing the periodic nature of the sine function.

So, the solution to the equation is:

\[
x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the general solution for \( \sin(x) = -1 \)?
2. How do you solve trigonometric equations involving multiple angles, like \( \sin(2x) = 1 \)?
3. How does the period of the sine function affect its solutions?
4. What are the solutions to \( \cos(x) = 1 \) and how do they compare to \( \sin(x) = 1 \)?
5. How do transformations (like scaling) affect the graph of \( \sin(x) \)?

**Tip:** Remember that trigonometric functions are periodic, meaning their values repeat at regular intervals, which is crucial when solving equations involving these functions.

Learn how to solve the trigonometric equation 6sin(x) = 6, where simplifying leads to the equation sin(x) = 1. Discover the general solution, which involves understanding the periodicity of the sine function. Suitable for students in Grades 9-11.

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