To solve the equation \( 2\sin(x) = \cos(x) \):

1. **Rewrite the equation**: Start by dividing both sides by \(\cos(x)\), assuming \(\cos(x) \neq 0\).
   \[
   \frac{2\sin(x)}{\cos(x)} = \frac{\cos(x)}{\cos(x)}
   \]
   This simplifies to:
   \[
   2\tan(x) = 1
   \]

2. **Solve for \(\tan(x)\)**:
   \[
   \tan(x) = \frac{1}{2}
   \]

3. **Find the general solution**:
   To solve \(\tan(x) = \frac{1}{2}\), we take the arctangent of both sides:
   \[
   x = \tan^{-1}\left(\frac{1}{2}\right) + k\pi, \quad k \in \mathbb{Z}
   \]
   Using the approximate value \(\tan^{-1}\left(\frac{1}{2}\right) \approx 0.4636\), the general solution is:
   \[
   x \approx 0.4636 + k\pi, \quad k \in \mathbb{Z}
   \]

Therefore, the solution to the equation \( 2\sin(x) = \cos(x) \) is:
\[
x \approx 0.4636 + k\pi, \quad k \in \mathbb{Z}
\]

Would you like more details or have any questions?

Here are some related questions:
1. How can we solve trigonometric equations with multiple trigonometric functions?
2. What are some methods to simplify trigonometric expressions?
3. How do we determine solutions for equations involving both sine and cosine?
4. What is the general solution for an equation involving \(\tan(x)\)?
5. How can the unit circle help in solving trigonometric equations?

**Tip:** When solving trigonometric equations, consider dividing by a trigonometric function only if you’re sure it’s not zero, as dividing by zero is undefined.

Learn how to solve the trigonometric equation 2sin(x) = cos(x) by converting it into a tangent equation. This detailed solution explores key concepts like the tangent function and inverse tangent to find the general solution. Suitable for students in Grades 10-12.

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