Which of the following pairs of polar coordinates represents the same point as $\left(3, \frac{\pi}{3}\right)$, such that $r < 0$ and $0 \le \theta < 2\pi$?

A) $\left(-3, \frac{4\pi}{3}\right)$

B) $\left(-3, \frac{\pi}{3}\right)$

C) $\left(-3, \frac{2\pi}{3}\right)$

D) $\left(-3, \frac{5\pi}{3}\right)$

Which of the following represents the polar coordinates $ (r, \theta) $, with $ r > 0 $ and $ 0 \le \theta < 2\pi $, of the point whose Cartesian coordinates are $ (-3, 3\sqrt{3}) $?

A) $ \left(6, \frac{\pi}{3}\right) $

B) $ \left(6, \frac{5\pi}{6}\right) $

C) $ \left(6, \frac{2\pi}{3}\right) $

D) $ \left(6, \frac{4\pi}{3}\right) $

Math · Functions & Trigonometry · Math · Polar, Parametric & Vectors

5 Math practice questions on Polar coordinates. Questions that assess core concepts and understanding.

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1.
What is the distance between the points with polar coordinates $P\left(2, \frac{\pi}{3}\right)$ and $Q\left(4, \frac{2\pi}{3}\right)$ ?
- A) $2\sqrt{7}$
- B) 4
- C) $2\sqrt{3}$
- D) 6
2.
Which of the following pairs of polar coordinates represents the same point as $\left(3, \frac{\pi}{3}\right)$ , such that $r < 0$ and $0 \le \theta < 2\pi$ ?
- A) $\left(-3, \frac{4\pi}{3}\right)$
- B) $\left(-3, \frac{\pi}{3}\right)$
- C) $\left(-3, \frac{2\pi}{3}\right)$
- D) $\left(-3, \frac{5\pi}{3}\right)$
3.
Which of the following is the Cartesian equation of the curve given by the polar equation $r = 6\sin\theta$ ?
- A) $x^2 + (y - 3)^2 = 9$
- B) $(x - 3)^2 + y^2 = 9$
- C) $x^2 + (y + 3)^2 = 9$
- D) $x^2 + y^2 = 36$
4.
Which of the following represents the polar coordinates $(r, \theta)$ , with $r > 0$ and $0 \le \theta < 2\pi$ , of the point whose Cartesian coordinates are $(-3, 3\sqrt{3})$ ?
- A) $\left(6, \frac{\pi}{3}\right)$
- B) $\left(6, \frac{5\pi}{6}\right)$
- C) $\left(6, \frac{2\pi}{3}\right)$
- D) $\left(6, \frac{4\pi}{3}\right)$
5.
Which of the following represents the rectangular (Cartesian) coordinates $(x, y)$ of the point given in polar coordinates as $\left(4, \frac{5\pi}{6}\right)$ ?
- A) $(2\sqrt{3}, -2)$
- B) $(-2, 2\sqrt{3})$
- C) $(2, -2\sqrt{3})$
- D) $(-2\sqrt{3}, 2)$

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