Problem 17 - Precalculus

Question

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(-3,-1.57)$$

Step-by-Step Solution

Answer

The given point can be plotted using the polar representation (-3, -1.57), and the two additional polar representations of this point are (-3, 4.71) and (3, 1.57).

Step 1: Plot the Point

The coordinates are given in polar form, that is (-3, -1.57). In polar coordinates, the first number is the radius (r) and the second is the angle (θ) in radians. Here, r is -3 and θ is -1.57. To plot this point in polar coordinates, one draws an angle of -1.57 radians, and then measures a distance of -3 units along this angle. Note that negative radii imply moving in the opposite direction.

Step 2: Find Additional Polar Representations

Any point in polar coordinates can be represented in many ways because adding 2π radians to θ doesn't change the position of the point, essentially creating a loop around the origin.For the first additional representation, θ can be increased by 2π: -1.57+2π = 4.71 while the r value stays the same. So, our first representation will be (-3,4.71).For the second additional representation, flip the direction of the radial line by changing the sign of the radius, r from -3 to 3, and opposing the angle θ, from -1.57 to -1.57+π = 1.57. This will end at the point (3, 1.57).

Step 3: Validation

After finding the additional polar representations, verify if all three points plot to the same location by using the conversion formulas x=r*cos(θ) and y=r*sin(θ), where r is the radius and θ is the angle. This will ensure whether or not the coordinates represent the same point.

Key Concepts

Angle Conversion

Understanding angle conversion is essential in polar coordinates. The angle specified, known as \( \theta \), determines the direction from the origin. Doing conversions often means adjusting this angle to fit specific conditions or constraints. Here, we have \(-1.57\) radians, roughly equivalent to \(-90^{\circ}\), and we need additional representations within the range \(-2\pi < \theta < 2\pi\).

A great trick when working with angles in polar coordinates is to remember that adding \(2\pi\) radians (or 360 degrees) will bring you back to the same orientation. So, adding \(2\pi\) to the original angle, we find \(-1.57 + 2\pi = 4.71\) radians, which is another way to describe that same direction.

If the concept of radians is new to you, visualize it as the distance along the circumference of a unit circle. Just remember:

• Conveniently, \(\pi\) radians is half a circle, equivalent to 180 degrees.
• Similarly, \(2\pi\) radians make a full circle, equivalent to 360 degrees.

Negative Radius

The concept of a negative radius can initially be tricky. In polar coordinates, negative radii essentially mean moving in the opposite direction at the specified angle. So, for a point with \(r = -3\), \(\theta = -1.57\), you go 3 units in the opposite direction of the specified angle, effectively reversing the direction.

Handling a negative radius involves flipping the angle. This means if you change your angle by \(\pi\) (or 180 degrees), you will point to the same line placed in the opposite direction. That's why, in our problem, flipping the radius from \(-3\) to \(3\) involves a change of our angle from \(-1.57\) to \(-1.57 + \pi = 1.57\).

Here's how to think about it:

• A negative radius changes your vector direction.
• Think of it as walking backward on the line defined by your angle.

Multiple Representations

Polar coordinates have a beautiful flexibility allowing multiple representations of the same point.

This flexibility comes from the nature of circular motion. Adding \(2\pi\) results in a full loop around the circle's origin, which means the angle representation is effectively the same point.

For our example \((-3, -1.57)\), we found multiple representations:

• By adding \(2\pi\), the angle becomes \(4.71\), maintaining the radius at \(-3\): \((-3, 4.71)\).
• By flipping the radius sign and adjusting angle by \(\pi\), we find the further representation as \((3, 1.57)\).

Make sure any new coordinate plots to the same location on the polar graph. Verify using the conversion into Cartesian coordinates:\[x = r \cos(\theta)\] and \[y = r \sin(\theta)\] Once validated, these alternative coordinates verify that although the forms differ, the location remains constant.