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To solve the equation \(\sin(\theta) = \cos(\theta)\) for the range \(0^\circ \leq \theta \leq 540^\circ\), we need to find all values of \(\theta\) where the sine and cosine functions are equal.

### Step-by-Step Solution:

1. **Understanding the equation:**

   \[
   \sin(\theta) = \cos(\theta)
   \]

   This equation holds true when the angle \(\theta\) forms a 45-degree reference angle in the unit circle. 

2. **Using the trigonometric identity:**

   \[
   \sin(\theta) = \cos(\theta) \implies \tan(\theta) = 1
   \]

   This is because \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), and for the tangent to equal 1, the angle must satisfy \(\tan(\theta) = 1\).

3. **Finding the general solutions:**

   The tangent function equals 1 at:
   \[
   \theta = 45^\circ + 180^\circ \cdot n, \quad \text{where } n \in \mathbb{Z}.
   \]

4. **Determining the specific solutions within the range:**

   Now, we find all values of \(\theta\) within the range \(0^\circ \leq \theta \leq 540^\circ\):

   - For \(n = 0: \theta = 45^\circ\)
   - For \(n = 1: \theta = 45^\circ + 180^\circ = 225^\circ\)
   - For \(n = 2: \theta = 225^\circ + 180^\circ = 405^\circ\)

   These are all within the range \(0^\circ \leq \theta \leq 540^\circ\).

5. **List of solutions:**

   The values of \(\theta\) that satisfy the equation \(\sin(\theta) = \cos(\theta)\) within the given range are:
   \[
   \theta = 45^\circ, \; 225^\circ, \; 405^\circ.
   \]

### Conclusion:

The values of \(\theta\) that satisfy the equation \(\sin(\theta) = \cos(\theta)\) for the range \(0^\circ \leq \theta \leq 540^\circ\) are:

- \(45^\circ\)
- \(225^\circ\)
- \(405^\circ\)

Would you like to know more details about the trigonometric identities involved, or have any other questions?

### Related Questions:

1. What are the solutions to the equation \(\sin(\theta) = -\cos(\theta)\)?
2. How do you find the general solution for \(\sin(x) = \cos(x)\) in radians?
3. What other angles have the property that \(\sin(\theta) = \cos(\theta)\)?
4. How can you use the unit circle to verify these solutions visually?
5. What are some practical applications of knowing when \(\sin(\theta) = \cos(\theta)\)?

#### Tip:

To solve trigonometric equations effectively, it is useful to remember key angle values on the unit circle and their corresponding sine, cosine, and tangent values.

Which of the following values satisfies the equation sin(θ) = cos(θ) for the range of 0° ≤ θ ≤ 540°?

Explore the solution to the trigonometric equation sin(θ) = cos(θ) within the range 0° ≤ θ ≤ 540°. This problem covers key trigonometric concepts and identities, including the tangent identity, with detailed step-by-step solutions. Suitable for high school students in Grades 10-12.

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