find a cartesian equation for the curve theta=-pi/3

To find the Cartesian equation for the polar equation \(\theta = -\frac{\pi}{3}\), follow these steps:

### Polar to Cartesian Conversion
1. The polar equation describes a straight line at a constant angle \(\theta = -\frac{\pi}{3}\).
2. In polar coordinates, the relation between polar and Cartesian coordinates is:
   \[
   x = r \cos(\theta)
   \]
   \[
   y = r \sin(\theta)
   \]
   
   where \(r\) is the radial distance and \(\theta\) is the angle.

### For \(\theta = -\frac{\pi}{3}\):
At this specific angle, the line makes an angle of \(-\frac{\pi}{3}\) radians (which is \(-60^\circ\)) with the positive x-axis.

The direction of this line can be described by the tangent of the angle:
\[
\tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}
\]
Thus, the equation of the line in Cartesian coordinates is:
\[
y = \tan\left(-\frac{\pi}{3}\right) x = -\sqrt{3}x
\]

### Final Cartesian Equation:
\[
y = -\sqrt{3}x
\]

This is the equation of the straight line corresponding to the polar angle \(\theta = -\frac{\pi}{3}\).

Would you like more details on this? Here are 5 related questions:

1. How do you derive the tangent of a specific angle?
2. What is the geometric interpretation of a constant \(\theta\) in polar coordinates?
3. Can you generalize the method to find a Cartesian equation for any \(\theta\)?
4. What if the radial distance \(r\) was not constant?
5. How do polar and Cartesian coordinates relate in three dimensions?

**Tip:** When converting between coordinate systems, it's useful to remember key angle values and their corresponding trigonometric functions.

Find a Cartesian equation for the curve theta = -pi/3.

Learn how to convert the polar equation θ = -π/3 into a Cartesian equation. This solution involves understanding polar and Cartesian coordinates, using trigonometric functions, and deriving a linear equation y = -√3x. Suitable for students in Grades 10-12.

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