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To convert the polar equation \( \theta = -\frac{3\pi}{4} \) into rectangular coordinates, we need to use the relationship between polar and rectangular coordinates:
\[
x = r \cos(\theta), \quad y = r \sin(\theta), \quad \text{and} \quad \theta = \arctan\left(\frac{y}{x}\right).
\]

Given \( \theta = -\frac{3\pi}{4} \), we can rewrite this using the rectangular form:
\[
\arctan\left(\frac{y}{x}\right) = -\frac{3\pi}{4}.
\]

To eliminate the arctan, we use the fact that:
\[
\tan\left(-\frac{3\pi}{4}\right) = \frac{y}{x}.
\]

Since \( \tan\left(-\frac{3\pi}{4}\right) = 1 \), we obtain:
\[
\frac{y}{x} = 1.
\]

Thus, the equivalent equation in rectangular coordinates is:
\[
y = x.
\]

Would you like more details, or do you have further questions?

Here are five related questions for further exploration:
1. How do you convert polar equations to rectangular form if the equation involves \( r \)?
2. What are the general relationships between polar and rectangular coordinates?
3. How would you convert \( r = 2 \sin(\theta) \) into rectangular form?
4. How can you graphically represent the equation \( \theta = -\frac{3\pi}{4} \)?
5. How does the tangent function behave at various angles?

**Tip:** When converting between coordinate systems, always keep the fundamental relationships (like \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \)) in mind for accurate conversions.

Convert the equation \( \theta = -\frac{3\pi}{4} \) into rectangular coordinates.

Learn how to convert the polar equation \( \theta = -\frac{3\pi}{4} \) into rectangular coordinates using trigonometric identities and algebraic methods. This solution is ideal for high school students in Grades 10-12.

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