Problem 42 - Precalculus : Building Concepts and Connections

Question

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(-2.7, \frac{5 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The additional pair of coordinates with \(r>0\) is \((2.7, \frac{\pi}{4})\) while for \(r<0\), it is \((-2.7, \frac{5\pi}{4})\).

Step 1: Get a pair of polar coordinates with \(r>0\)

Starting from the given point \((-2.7, \frac{5\pi}{4})\), let us find a pair of polar coordinates with positive \(r\). To do this, we add \(\pi\) to the current angle. We can write \(r = -2.7\) and \(\theta = \frac{5\pi}{4} + \pi = \frac{9\pi}{4}\). However, as \(\theta\) could be reduced by subtracting \(2\pi\) we get \(\theta = \frac{\pi}{4}\) which gives us the new polar coordinates \((2.7, \frac{\pi}{4})\).

Step 2: Get a pair of polar coordinates with \(r<0\)

From the given point \((-2.7, \frac{5\pi}{4})\), let us get a different pair of polar coordinates, but in this case, with negative \(r\). To do this, subtract \(\pi\) from the current angle. So, \(r = 2.7\) (as we established in the previous step) and \(\theta = \frac{\pi}{4} - \pi = -\frac{3\pi}{4}\). To keep the angle positive, we add \(2\pi\) to get \(\theta = \frac{5\pi}{4}\) which results in our new polar coordinates \((-2.7, \frac{5\pi}{4})\), bringing us back to the original pair.

Key Concepts

Coordinate Transformation

Polar coordinates are an alternative way to represent points in a plane, especially useful in systems where the relationship of points is more naturally expressed in terms of angles and distances from a central point. Unlike the Cartesian coordinate system that uses x and y to define points on a plane, polar coordinates define a point using a radius, \(r\), which is the distance from the origin, and an angle, \(\theta\), measured counterclockwise from the positive x-axis.
To transform a point from polar to Cartesian coordinates, we use the formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

For instance, the point given as \((-2.7, \frac{5\pi}{4})\) in polar coordinates describes a point that is 2.7 units away from the origin, making an angle of \(\frac{5\pi}{4}\) with the positive x-axis, but on the opposite side due to the negative radius.

Angle Conversion

Angles in polar coordinates can sometimes appear in unconventional forms that need adjustment for practical or specific applications. The angle \(\theta\) can be modified by adding or subtracting full rotations (\(2\pi\) radians), which doesn't change the direction.
A key transformation involves ensuring the radius \(r\) is positive, which entails converting the angle \(\theta\) by altering it by \(\pi\). This is useful because polar coordinates are periodic, repeating every \(2\pi\) radians, allowing for multiple representations of the same point.
In the exercise, modifying \((-2.7, \frac{5\pi}{4})\) to a form where \(r > 0\) involves adjusting the angle to \(\frac{\pi}{4}\) by adding \(\pi\) (or subtracting \(-\pi\)), and then simplifying to keep angles within the principal range (\(0\) to \(2\pi\)).

Trigonometric Concepts

Trigonometric concepts play a crucial role in understanding polar coordinates. The conversion between polar and Cartesian coordinates relies heavily on the sine and cosine functions, representing projections of the radius onto the x and y axes, respectively.
Using trigonometric identities, we can explain why different angles can represent the same direction. The symmetry of sine and cosine functions under rotation is key to solving these problems, such as adding \(\pi\) to ascertain opposite directions.
For example, in our exercise, by using the identity \(\cos(\theta + \pi) = -\cos(\theta)\) and \(\sin(\theta + \pi) = -\sin(\theta)\), we discover why adding or subtracting \(\pi\) in a polar coordinate fundamentally changes the direction from one positive or negative radial value to its opposite. Understanding these relationships is vital for grasping the nuances of polar coordinate transformations.