what values satisfy the equation sin(theta) =cos(theta) for the range of 0 less than equal to theta less than equal to 720

The equation given is:

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

To solve this, we can use the identity:

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

This simplifies to:

\[
\theta = \frac{\pi}{4} + n\pi
\]

Where \( n \) is any integer.

Since the problem asks for solutions within the range \( 0 \leq \theta \leq 720^\circ \), we need to find all values of \( \theta \) in degrees within this range.

### Step-by-step Calculation:

1. **Starting with the general solution:**
   \[
   \theta = \frac{\pi}{4} + n\pi
   \]
   Convert to degrees:
   \[
   \theta = 45^\circ + 180^\circ \times n
   \]

2. **Find specific solutions:**
   - For \( n = 0 \): \(\theta = 45^\circ\)
   - For \( n = 1 \): \(\theta = 225^\circ\)
   - For \( n = 2 \): \(\theta = 405^\circ\)
   - For \( n = 3 \): \(\theta = 585^\circ\)

These are the solutions within the given range:

\[
\theta = 45^\circ, 225^\circ, 405^\circ, 585^\circ
\]

### Final Answer:
The values of \( \theta \) that satisfy the equation \( \sin(\theta) = \cos(\theta) \) within the range \( 0 \leq \theta \leq 720^\circ \) are:
\[
\theta = 45^\circ, 225^\circ, 405^\circ, 585^\circ
\]

Would you like further details on the calculation, or do you have any questions?

### Related Questions:
1. What is the general solution for \( \tan(\theta) = 1 \) in radians?
2. How does the period of the sine and cosine functions affect the number of solutions?
3. What is the graphical interpretation of the equation \( \sin(\theta) = \cos(\theta) \)?
4. Can you explain how to solve \( \sin(\theta) = \cos(\theta) \) using the unit circle?
5. What happens if the range is extended beyond \( 720^\circ \)?

**Tip:** Remember that the sine and cosine functions have a period of \( 360^\circ \), which means the solutions to the equation repeat every \( 180^\circ \).

What values satisfy the equation sin(theta) = cos(theta) for the range of 0 less than equal to theta less than equal to 720?

Explore how to solve the trigonometric equation sin(theta) = cos(theta) within the range 0° ≤ θ ≤ 720°. This problem involves key trigonometric concepts, including the use of trigonometric identities and general solutions. Suitable for students in Grades 10-12.

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