Question

Give two alternative representations of the points in polar coordinates. $$\left(-4, \frac{3 \pi}{2}\right)$$

Short answer

Answer: The two alternative representations of the point in polar coordinates are \((-4, \frac{7\pi}{2})\) and \((4, \frac{\pi}{2})\).

Step-by-step solution

Identify the given polar coordinates

The given polar coordinates are \((-4, \frac{3\pi}{2})\). We have \(r = -4\) and \(\theta = \frac{3\pi}{2}\).

Add or subtract multiples of \(2\pi\) to the given angle

We can add or subtract multiples of \(2\pi\) to the angle without changing the location of the point. In this case, let's add \(2\pi\) to the given angle: New angle: $$\theta_1 = \frac{3 \pi}{2} + 2 \pi = \frac{3 \pi}{2} + \frac{4 \pi}{2} = \frac{7 \pi}{2}$$ Alternative representation 1: \((-4, \frac{7\pi}{2})\)

Negate the distance and add or subtract odd multiples of \(\pi\) from the given angle

We can negate the distance and add or subtract odd multiples of \(\pi\) from the angle to find another alternative representation: New distance: $$r_1 = -(-4) = 4$$ New angle (subtracting \(\pi\)): $$\theta_2 = \frac{3 \pi}{2} - \pi = \frac{3 \pi}{2} - \frac{2 \pi}{2} = \frac{ \pi}{2}$$ Alternative representation 2: \((4, \frac{\pi}{2})\) The two alternative representations of the point in polar coordinates are: $$\left(-4, \frac{7\pi}{2}\right) \text{ and } \left(4, \frac{\pi}{2}\right)$$

Key concepts

Angle Addition in Polar Coordinates

In the realm of polar coordinates, adding or subtracting certain angles can help represent the same point differently. This is because angles in polar coordinates are periodic. You might wonder why we can just add angles and not change the point's position at all. The key lies in recognizing that a full circle, or complete rotation, measures \(2\pi\) radians. Therefore, \(2\pi\) radians added or subtracted to an angle brings the point back to the initial heading, since it's just a full loop around the circle.

• For instance, given an angle \(\frac{3\pi}{2}\), adding \(2\pi\) gets us to \(\frac{7\pi}{2}\).
• The coordinates \((-4, \frac{3\pi}{2})\) change to \((-4, \frac{7\pi}{2})\) upon angle addition.

This is a useful tool in polar coordinate transformation as it maintains the point's location but provides a different visual interpretation, which can aid in solving problems by offering alternative views.

Coordinate Transformation: Polar to Polar

Coordinate transformation involves changing how coordinates are represented, without altering the point's position itself. In polar coordinates, this can mean altering either the direction by adjusting the angle, or the magnitude by changing the radius, or both.

• One way is through angle addition, as previously discussed.
• Another method is through distance negation, which takes a bit of a different path.

When it comes to transforming coordinates, the focus is often on changing values like the angle for clarity in problems. Still, it holds the essence of keeping the point constant in its original or intended position without actual movement in space. This allows students to understand better how each mathematical change affects or reroutes understanding while staying true to the origin of the coordinates.

Understanding Distance Negation

Distance negation is an intriguing concept in polar coordinates that helps to find alternative representations by reversing the sign of the radius. In simple terms, negating the distance ( ") means switching sides of the origin polar point.

• If the radius is negative, like -4, negating it becomes positive ( = 4).
• To achieve this without shifting the point, altering the angle by \(\pi\) or other odd multiples is necessary.

Through this, you maintain the directionality, essentially mirroring the point over the origin. By flipping it around and changing the perspective with a short angle adjustment, the new coordinates remain linked to their original as valid representations in a different frame. Thus, a point like \((-4, \frac{3\pi}{2})\) can be transformed to \(4, \frac{\pi}{2}\). This is an action that mathematics uses to maintain coherence and flexibility in problem-solving, allowing students to see how changes affect viewpoints without altering the position.