Finding an equivalent polar coordinate representation

Key concept: This question checks the two standard rules for polar coordinates: adding multiples of $2\pi$ and switching the sign of $r$.

Step by step

Key idea

A polar point can be represented in more than one way. The two main equivalence rules are:

1. $(r,\theta)\equiv(r,\theta+2k\pi)$
2. $(r,\theta)\equiv(-r,\theta+(2k+1)\pi)$

Here the given point is

$(-3,\tfrac{2\pi}{3}).$

Because the radius is negative, we can rewrite the point with a positive radius by adding $\pi$ to the angle:

$(-3,\tfrac{2\pi}{3})\equiv(3,\tfrac{2\pi}{3}+\pi).$

Now simplify the angle:

$\tfrac{2\pi}{3}+\pi=\tfrac{2\pi}{3}+\tfrac{3\pi}{3}=\tfrac{5\pi}{3}.$

So the equivalent polar coordinate is

$\boxed{(3,\tfrac{5\pi}{3})}$

which is choice A.

Why this works

A negative radius means you move in the direction opposite the terminal ray of the angle. Flipping the sign of $r$ requires a half-turn, which is exactly $\pi$ radians. That is why the angle changes by $\pi$ when $r$ changes sign.

If you only add $2\pi$ to the angle while keeping $r=-3$, the location is still the same point, but the answer choices ask for another representation, and the positive-radius form is the clearest match.

Check the choices

• A $(3,\frac{5\pi}{3})$: correct, because it uses the same point with positive radius.
• B $(3,\frac{2\pi}{3})$: incorrect, because changing the sign of $r$ without changing the angle changes the point.
• C $(3,\frac{\pi}{3})$: incorrect, because this is a different terminal ray.
• D $(-3,\frac{7\pi}{3})$: equivalent to the original point, but the intended standard equivalent form is still choice A.

So the correct answer is $\boxed{A}$.

Pitfall alert

A frequent mistake is to think that changing $r$ from negative to positive does not affect the angle. In polar coordinates, a negative radius always points in the opposite direction, so the angle must shift by $\pi$. Another error is adding $2\pi$ and expecting that alone to change a negative-radius point into a positive-radius point. It does not. The sign change and the angle shift work together. Also watch the trigonometric quadrant: $\frac{5\pi}{3}$ is in Quadrant IV, which matches the direction opposite $\frac{2\pi}{3}$.

Try different conditions

If the point were $(3,\frac{2\pi}{3})$, then an equivalent representation would be $(-3,\frac{5\pi}{3})$, because switching the sign of $r$ requires adding $\pi$ to the angle. If the point were $(-3,\frac{7\pi}{3})$, you could subtract $2\pi$ to rewrite it as $(-3,\frac{\pi}{3})$, which is still the same location. These variants show that either the radius sign changes with a $\pi$ shift, or the angle changes by multiples of $2\pi$.

Further reading

polar coordinate equivalence, negative radius, terminal ray