Rewrite the following polar equation in rectangular form. Write your answer in exact form.
 

θ=−π3


To rewrite the polar equation \(\theta = -\frac{\pi}{3}\) in rectangular form, we need to use the relationship between polar and rectangular coordinates.

Polar coordinates \((r, \theta)\) can be converted to rectangular coordinates \((x, y)\) using the following formulas:
\[
x = r \cos \theta
\]
\[
y = r \sin \theta
\]

Given \(\theta = -\frac{\pi}{3}\), we need to find the line that corresponds to this angle. In polar coordinates, \(\theta = -\frac{\pi}{3}\) represents all points that form an angle of \(-\frac{\pi}{3}\) with the positive \(x\)-axis.

The slope \(m\) of the line is given by:
\[
m = \tan \theta = \tan\left(-\frac{\pi}{3}\right)
\]

Using the tangent of \(-\frac{\pi}{3}\):
\[
\tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}
\]

Therefore, the line has a slope of \(-\sqrt{3}\). In slope-intercept form (\(y = mx + b\)), since the line passes through the origin (for any \(r\)), the intercept \(b\) is zero.

Thus, the equation of the line is:
\[
y = -\sqrt{3} x
\]

This is the rectangular form of the given polar equation \(\theta = -\frac{\pi}{3}\).

Would you like more details or have any questions?

Here are 8 related questions that expand on this information:
1. How do you convert a polar equation involving \(r\) and \(\theta\) to rectangular form?
2. What is the significance of the angle \(\theta\) in polar coordinates?
3. How would you convert the polar equation \(r = 2\cos \theta\) to rectangular form?
4. What are the polar coordinates of the point \((1, -\sqrt{3})\)?
5. How do you find the slope of a line given in polar coordinates?
6. Can a polar equation have more than one rectangular form?
7. How would you convert the rectangular equation \(x^2 + y^2 = 9\) to polar form?
8. What is the general approach to solving \(\theta\) for a given line equation in polar coordinates?

**Tip:** Remember that angles in polar coordinates are measured from the positive \(x\)-axis, with positive angles measured counterclockwise and negative angles measured clockwise.

Learn how to rewrite the polar equation θ = -π/3 into rectangular form using trigonometric functions and conversion formulas. This problem explores the relationship between polar and rectangular coordinates, focusing on the tangent function for slope determination. Suitable for college-level mathematics students.

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