Question

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

Short answer

Answer: The two alternative representations for the polar coordinate (-1, -π/3) are (1, 8π/3) and (1, -4π/3). The corresponding rectangular coordinates for both alternative representations are (-1/2, √3/2).

Step-by-step solution

Convert the polar coordinate to a positive radius and positive angle

To change the negative radius to a positive radius, we add π to the angle and take the absolute value of the radius. So, we have: \(r = |-1| = 1\) and \(\theta = -\frac{\pi}{3} + \pi = \frac{2\pi}{3}\) Thus, our new polar coordinate is \((1, \frac{2\pi}{3})\).

Convert the new polar coordinate to rectangular form

To convert the polar coordinates to rectangular coordinates, we use the relationships: \(x = r\cos\theta\) and \(y = r\sin\theta\) Plugging in the values we have: \(x = 1\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\) and \(y = 1\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\) Thus, our rectangular coordinate is \((-\frac{1}{2}, \frac{\sqrt{3}}{2})\).

Find two alternative representations

We can find two alternative representations of the point by adding and subtracting 2π to the angle in our new polar coordinate. Alternative 1: \((1, \frac{2\pi}{3} + 2\pi) = (1, \frac{8\pi}{3})\) Alternative 2: \((1, \frac{2\pi}{3} - 2\pi) = (1, -\frac{4\pi}{3})\)

Convert alternative coordinates to rectangular form

For both alternative representations, due to the periodic nature of sine and cosine functions, they will have the same rectangular coordinates as our initial representation. Alternative 1: \(x = 1\cos\left(\frac{8\pi}{3}\right) = -\frac{1}{2}\), \(y = 1\sin\left(\frac{8\pi}{3}\right) = \frac{\sqrt{3}}{2}\) Thus, our rectangular coordinate is \((-\frac{1}{2}, \frac{\sqrt{3}}{2})\). Alternative 2: \(x = 1\cos\left(-\frac{4\pi}{3}\right) = -\frac{1}{2}\), \(y = 1\sin\left(-\frac{4\pi}{3}\right) = \frac{\sqrt{3}}{2}\) Thus, our rectangular coordinate is \((-\frac{1}{2}, \frac{\sqrt{3}}{2})\). The two alternative representations for the polar coordinate \((-1, -\frac{\pi}{3})\) are \((1, \frac{8\pi}{3})\) and \((1, -\frac{4\pi}{3})\).

Key concepts

Converting Polar Coordinates to Rectangular Coordinates

Understanding how to switch between polar and rectangular coordinates is a fundamental skill in mathematics, particularly in subjects like trigonometry, calculus, and physics.

Each point in a polar coordinate system is denoted by a pair \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle in radians. To convert this to rectangular coordinates \( (x, y) \) we use the defining relations: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). These formulas arise from the definition of sine and cosine in a right-angled triangle.

In the exercise provided, the point with polar coordinates \( (-1, -\frac{\pi}{3}) \) is converted to \( (1, \frac{2\pi}{3}) \) by adjusting for a positive radius, which effectively mirrors the point across the origin. Then, we apply the conversion formulas to obtain the rectangular coordinates \( (-\frac{1}{2}, \frac{\sqrt{3}}{2}) \) which represent the same point in the Cartesian plane.

Remember, the key is to be comfortable with the polar coordinate system and the relationship between the trigonometric functions and the right triangle formed by the radius and the axes.

Polar and Rectangular Coordinate Systems

Two common coordinate systems used in mathematics and science are the polar and rectangular (or Cartesian) systems. Each system has its own situation where it is most useful.

In the polar coordinate system, points are determined by how far away they are from the origin, denoted as \( r \), and the angle they make with the positive x-axis, represented by \( \theta \). This system is particularly useful for problems involving symmetry around a central point or when dealing with phenomena that naturally exhibit radial symmetry, like waves emanating from a point.

On the other hand, the rectangular coordinate system uses two perpendicular lines, the x and y axes, to define a point in space. This system is typically the first one learned and is adept at dealing with linear equations and defining rectangles, hence the term rectangular coordinates.

Understanding how to convert between these two systems can enhance a student's versatility in solving various geometric and trigonometric problems.

Periodicity of Trigonometric Functions

A unique property of trigonometric functions is their periodicity, meaning they repeat their values in regular intervals. For instance, the sine and cosine functions have a period of \( 2\pi \) radians, which means that adding or subtracting multiples of \( 2\pi \) to the angle \( \theta \) will not change the value of these functions.

In our exercise, alternative representations for the polar coordinate are found by adding or subtracting \( 2\pi \) because of this periodic nature. The alternatives \( (1, \frac{8\pi}{3}) \) and \( (1, -\frac{4\pi}{3}) \) essentially map back to the original point on the Cartesian plane due to the functions repeating their values every \( 2\pi \) interval. That's why their rectangular coordinates are the same as the initial conversion.

Appreciating the concept of periodicity allows students to tackle problems involving trigonometric functions more effectively and recognize patterns in related graphs. Periodicity is an essential concept in fields such as signal processing and physics, as well as in the study of waves and oscillations.