Problem 10 - Precalculus

Question

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(4,5 \pi / 2)$$

Step-by-Step Solution

Answer

The two equivalent polar representations of the point \( (4,5\pi/2) \) within the range \(-2\pi < \theta < 2\pi\) are \( (4, \pi/2) \) and \( (4, -3\pi/2) \).

Step 1: Plot the point

Plotting the point represented by the given polar coordinates is the first step. Start at the origin and move 4 units along the reference line and then rotate \(5 \pi / 2\) radians anticlockwise. Note that \(5 \pi / 2\) is equivalent to \(2\pi + \pi/2\), which is a full rotation plus a quarter rotation. Hence, the point lies on the y-axis, 4 units above the origin.

Step 2: Determine an equivalent polar representation

The point \(4,5\pi/2)\) can have equivalent polar representations by adding or subtracting multiples of \(2\pi\) to/from the angle. First, let's subtract \(2\pi\) (one full rotation) from the angle. This gives a representation of \( ( 4, 5\pi/2 - 2\pi ) = (4, \pi/2 ) \). This point is within the range \(-2\pi < \theta < 2\pi\). Such subtraction does not affect the location of the point because it simply denotes one less full rotation.

Step 3: Determine a second equivalent polar representation

Next, subtract another \(2\pi\) from the angle part of the coordinates. This results in \( ( 4, \pi/2 - 2\pi ) = (4, - 3\pi/2) \). This is the second polar representation within the specified range \(-2\pi < \theta < 2\pi\).

Key Concepts

Polar Representation

In polar coordinates, a point in the plane is represented by a pair

• r, which is the radial distance from the origin
• θ, which is the angle measured from the positive x-axis

The representation (r, θ) specifies • how far to travel from the origin, and • how much to rotate. Comparing this to Cartesian coordinates (x, y), the emphasis is on direction and distance.

An important thing to note is that each point has infinitely many representations. By adding or subtracting multiples of \(2\pi\) (a full circle rotation) to \(θ\), we do not change the point's position on the plane.

Angle Conversion

Converting angles between different forms or within certain confines is crucial for understanding and simplifying polar coordinates.

When finding alternate representations, knowing that \(\pi\) radians equals 180 degrees helps. In the exercise, the angle \(5\pi/2\) needs adjustment to fit the range requirement \(-2\pi < \theta < 2\pi\).

• \(5\pi/2\) is more than one full circle. Splitting it into \(2\pi + \pi/2\), we simplify it to \(\pi/2\)
• Further subtraction of \(2\pi\) gives \(-3\pi/2\), placing the angle within the desired range

Understanding these conversions allows accurate interpretations and manipulations of polar representations.

Plotting Points in Polar Form

Plotting points in polar form involves two main actions:

• Moving r units radially from the origin
• Rotating by θ to find the correct angle

For \( (4, 5\pi/2) \):

• First, it moves 4 units from the origin.
  
• Next, rotates \(5\pi/2\) radians (over 360 degrees), simplifying to \(\pi/2\), landing on the positive y-axis.

Changes in θ don’t alter its radial distance; this is crucial to finding those alternate representations successfully. Mastering this technique ensures precision in plotting any polar coordinate point.