Problem 15 - Precalculus with Limits

Question

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(\sqrt{2}, 2.36\right)\)

Step-by-Step Solution

Answer

The two additional polar representations of the point \(\left(\sqrt{2}, 2.36\right)\) are \(\left(\sqrt{2}, 8.78\right)\) and \(\left(\sqrt{2}, -3.92\right)\).

Step 1: Plot Original Point

Plot the point \(\left(\sqrt{2}, 2.36\right)\) in polar coordinates. The radius is \(\sqrt{2}\) and the angle is \(2.36\) radians.

Step 2: Find Additional Representations

To find additional representations of the point, remember that the angle to represent points in the polar system can be added or subtracted \(2π\). Additional representations are then found by adding and subtracting \(2π\) to the original angle of \(2.36\) radians. Thus, the first additional representation is \(\left(\sqrt{2}, 2.36+2π\right)\), and the second additional representation is \(\left(\sqrt{2}, 2.36-2π\right)\).

Step 3: Simplify for Final Representations

Simplify both the representations to get their final form. The first additional representation becomes \(\left(\sqrt{2}, 2.36+2π\right) = \left(\sqrt{2}, 8.78\right)\), and the second representation becomes \(\left(\sqrt{2}, 2.36-2π\right)=\left(\sqrt{2}, -3.92\right)\)

Key Concepts

Angle Measurement

In mathematics, particularly when discussing polar coordinates, understanding angle measurement is key. Angles determine the orientation of a point or an object in space.
Helps you know which direction you are pointing in a coordinate system.

There are three primary systems used to measure angles:

• Degrees
• Radians
• Gradians

The degree system might be more familiar, where a full circle is divided into 360 parts. This system is often used in day-to-day activities and makes it easy to relate angles to your environment. For example, a quarter turn is 90 degrees.
Radians are another way to measure angles, as you'll see later in this article. They're used more in mathematics due to their relationship with the radius of a circle. In radians, a full circle is measured as \(2π\), creating a direct link with the concept of a circle and its circumference. Understanding how to convert between these systems, especially between degrees and radians, can enhance your mathematical fluency and allow you to tackle various problems with greater ease.

Radian

A radian is an angle measurement based on the radius of a circle. When discussing mathematics, radians are perhaps the most natural unit for angle measure.
They link directly to the properties of circles and help simplify equations involving circular arcs.

In a full circle, there are \(2π\) radians. This means that one radian is directly tied to the radius of that circle. An important conversion to remember is that \(π\) radians equals 180 degrees.
To convert degrees to radians, you can use the formula:

• \(radians = degrees \times \frac{π}{180}\)

This conversion is particularly helpful when working on problems involving circles or periodic functions. In the polar coordinate exercise, noticing that an angle is given in radians helps determine how far along the circumference the point lies. Adding and subtracting \(2π\) gives additional representations because it represents one full cycle around the circle. With practice, using radians becomes second nature to solving geometric and trigonometric problems.

Coordinate System

A coordinate system is a framework used to uniquely identify points in space. It provides a method to pinpoint a location using numbers. There are different types of coordinate systems, each useful in various scenarios.
Some common coordinate systems include:

• Cartesian
• Polar
• Cylindrical
• Spherical

The polar coordinate system, used in your exercise, is particularly fascinating because it uses a radius and an angle for location identification. This system shines when dealing with scenarios involving circular shapes and rotations, as you have seen with the point \((\sqrt{2}, 2.36)\).
In polar coordinates:

• The first number, often noted as \(r\), indicates the distance from the origin, or pole.
• The second number, \(θ\), provides the direction based on an angle.

Exploring further into polar coordinates, you can see that different polar representations of the same point are possible by adding or subtracting \(2π\), allowing for multiple ways to express the same location. These nuances make understanding coordinate systems fundamental in both theoretical and applied mathematics. By mastering different systems, you can effectively navigate and solve a variety of mathematical challenges.