To solve the equation \( \sin(2\theta) = \cos(3\theta) \), we'll use trigonometric identities and algebraic manipulation.

### Step 1: Express both sides using a common trigonometric function
We know from trigonometric identities that:
\[
\cos(3\theta) = \sin\left(\frac{\pi}{2} - 3\theta\right)
\]
So the equation becomes:
\[
\sin(2\theta) = \sin\left(\frac{\pi}{2} - 3\theta\right)
\]

### Step 2: Use the sine identity
If \(\sin A = \sin B\), then:
\[
A = B + 2k\pi \quad \text{or} \quad A = \pi - B + 2k\pi \quad \text{for any integer } k
\]
Applying this to our equation:
\[
2\theta = \frac{\pi}{2} - 3\theta + 2k\pi
\]
or
\[
2\theta = \pi - \left(\frac{\pi}{2} - 3\theta\right) + 2k\pi
\]

### Step 3: Solve for \(\theta\)
#### Case 1:
\[
2\theta + 3\theta = \frac{\pi}{2} + 2k\pi
\]
\[
5\theta = \frac{\pi}{2} + 2k\pi
\]
\[
\theta = \frac{\pi}{10} + \frac{2k\pi}{5}
\]

#### Case 2:
\[
2\theta = \frac{\pi}{2} + 3\theta - \pi + 2k\pi
\]
\[
-5\theta = -\frac{\pi}{2} + 2k\pi
\]
\[
\theta = \frac{\pi}{10} + \frac{2k\pi}{5}
\]
Both cases give us the same solution:
\[
\theta = \frac{\pi}{10} + \frac{2k\pi}{5}
\]

Thus, the general solution for \(\theta\) is:
\[
\theta = \frac{\pi}{10} + \frac{2k\pi}{5} \quad \text{for any integer } k
\]

Would you like more details on any step or have any questions?

Here are some related questions to explore:
1. How would you solve if the equation was \( \sin(2\theta) = \sin(3\theta) \)?
2. What is the general solution to \( \cos(2\theta) = \sin(3\theta) \)?
3. How would the solution change if the equation was \( \tan(2\theta) = \tan(3\theta) \)?
4. How can you use trigonometric identities to simplify \( \sin(4\theta) + \sin(2\theta) \)?
5. What are the implications of periodicity in the solution of trigonometric equations?

**Tip:** Always check if there are multiple angles that satisfy a trigonometric equation, as trigonometric functions are periodic.

Learn how to solve the trigonometric equation sin(2θ) = cos(3θ) using identities and algebraic manipulation. This problem covers key trigonometric concepts and identities, suitable for students in Grades 10-12.

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