Question

Give two alternative representations of the points in polar coordinates. $$\left(-1,-\frac{\pi}{3}\right)$$

Short answer

Answer: The two alternative representations of the given point in polar coordinates are (-1, 5π/3) and (1, 11π/3).

Step-by-step solution

1. Add \(2\pi\) to the angle when the radius is negative

We start by adding \(2\pi\) to the angle -\(\frac{\pi}{3}\). This gives us: $$-\frac{\pi}{3} + 2\pi = -\frac{\pi}{3} + \frac{6\pi}{3} = \frac{5\pi}{3}$$ The point in polar coordinates, after adding \(2\pi\) to the angle, is: $$\left(-1,\frac{5\pi}{3}\right)$$

2. Change sign of the radius

Now, we want to change the sign of the radius and add a different multiple of \(2\pi\) to the angle. Let's first change the sign of the radius: $$(-1)\times(-1)=1$$ And now, add a different multiple of \(2\pi\), such as \(4\pi\), to the angle: $$-\frac{\pi}{3} + 4\pi= -\frac{\pi}{3} + \frac{12\pi}{3} = \frac{11\pi}{3}$$ The point in polar coordinates, after changing the sign of the radius and adding a different multiple of \(2\pi\) to the angle, is: $$\left(1,\frac{11\pi}{3}\right)$$ So, the two alternative representations of the given point in polar coordinates are: $$\left(-1,\frac{5\pi}{3}\right) \text{ and } \left(1,\frac{11\pi}{3}\right)$$

Key concepts

Understanding Trigonometry in Polar Coordinates

Trigonometry plays a vital role in understanding polar coordinates. In this system, points are represented using a radius and an angle from a fixed direction, usually the positive x-axis. Unlike Cartesian coordinates which use a grid system, trigonometry helps convert angles and distances into coordinates on a circle.
When dealing with polar coordinates, the angle is measured in radians, which comes from trigonometry. Radians are a way of expressing angles based on the radius of a circle. An important aspect to remember is that angles can be converted to their equivalent plus or minus multiples of \(2\pi\) radians, which constitutes a full circle, thanks to periodicity in trigonometric functions.

• Conversion from polar to Cartesian requires cosine and sine functions: - \(x = r \cos(\theta)\)
• The y-coordinate is similarly derived: - \(y = r \sin(\theta)\)

These trigonometric transformations help us visualize and comprehend how points in polar coordinates translate into flat surfaces that we understand with Cartesian coordinates.

Mathematical Transformations in Polar Coordinates

Mathematical transformations are significant in polar coordinates because they allow us to manipulate and express coordinates differently without changing their actual position in space.
When you alter the radius and angle, you're essentially performing a mathematical transformation. In the given example, transforming the point \((-1, -\frac{\pi}{3})\) involves two main steps. First, adjust the angle by adding \(2\pi\) or another multiple of it, reflecting the fact that the same angle can wrap around a circle indefinitely.
Changing the sign of the radius (from negative to positive or vice versa) is another transformation. This changes how far the point is from the origin but in the opposite direction. For instance, turning \(-1\) into \(1\) without changing the angle means we are interpreting the point in the opposite half of the circle. This flexibility in representation keeps the concept both interesting and practical. Transformations help ensure that every point can be described in different ways based on our needs and the problem context.

Diverse Coordinate Systems

Coordinate systems are frameworks that allow us to precisely define the position of points in space. Polar coordinates and Cartesian coordinates are two different systems that serve different purposes but can often be translated into each other.
Polar coordinates are especially useful in scenarios dealing with circular motion or phenomena with inherent symmetry around a point. Unlike Cartesian coordinates, which use \(x\) and \(y\) for specifying a point, polar coordinates involve a radius (\(r\)) and an angle (\(\theta\)) which express position in terms of distance from a central point and direction.

• Advantages of Polar Coordinates: - Simplifies mathematics in circular paths. - Enhances understanding of angular relationships.
• Conversions between systems can be done using trigonometric identities as bridges.

Each coordinate system comes with its own set of rules and ideal scenarios for use, making them versatile tools in mathematics. By learning to switch between these systems, you develop a more profound grasp of how spatial relationships can be interpreted and manipulated.