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To solve equations involving trigonometric functions, the goal is to find the angles, typically denoted as \( \theta \), that satisfy the given conditions. These steps will guide you through the process of solving trigonometric equations:

Step 1: **Rewrite the Equation in Standard Form**
The first step is to manipulate the original equation into a standard form such as zero on one side. For example, for \( 2 \sin^{2} \theta = \sin \theta \), we rewrite it as:

\[ 2 \sin^{2} \theta - \sin \theta = 0 \]

Step 2: **Factor the Equation**
Factoring allows you to split the equation into simpler parts. For instance:

\[ \sin \theta (2 \sin \theta - 1) = 0 \]

Step 3: **Set Each Factor to Zero**
Once factored, set each factor equal to zero and solve for \( \theta \):

\[ \sin \theta = 0 \quad \text{and} \quad 2 \sin \theta - 1 = 0 \]

This approach simplifies the solving process by breaking down the equation into more manageable pieces.

Factoring trigonometric expressions follows similar principles to factoring algebraic expressions and is essential for simplifying equations. Here’s a step-by-step guide to factoring trigonometric equations:

**Step 1: Identify Common Factors**
Look for terms that can be factored out, just like you would in algebra. For \( 2 \sin^{2} \theta - \sin \theta = 0 \), both terms include \( \sin \theta \):

\[ \sin \theta (2 \sin \theta - 1) = 0 \]

**Step 2: Apply the Zero Product Property**
Set each factor equal to zero:

\[ \sin \theta = 0 \quad \text{and} \quad 2 \sin \theta - 1 = 0 \]

**Step 3: Solve Each Equation**
Solve for \( \theta \) in each resulting equation separately. This method efficiently isolates potential solutions. For example:

\[ 2 \sin \theta = 1 \quad \rightarrow \quad \sin \theta = \frac{1}{2} \]

Factoring simplifies solving by reducing complex expressions into simple, solvable components.

Identifying valid solutions involves ensuring that solutions fit within the given constraints of the problem. For our equation, \(2 \sin ^{2} \theta=\theta, 0 < \theta \leq \pi\), certain steps are followed to ensure correctness:

**Step 1: Check Range Limitations**
Verify that the solved values of \( \theta \) fall within the specified range. For \( 0 < \theta \leq \pi \), values like \( \theta = 0 \) are excluded, while \( \theta = \pi \) is the edge case.

**Step 2: Validity of Solutions**
For \( \sin \theta = 0 \), within the specified range, the valid solution is \( \theta = \pi \).

**Step 3: Combine Valid Solutions**
List all angles that meet the conditions:

\[ \theta = \frac{\pi}{6}, \frac{5 \pi}{6}, \pi \]

Avoid missing solutions by scrutinizing how the problem constraints affect calculated values. Solving trigonometric equations accurately requires careful consideration of the defined range and any implied limitations from the initial manipulation of the equations.