Problem 17 - 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 \(-2\pi<\theta<2\pi\). \(\left(-3, -1.57\right)\)

Step-by-Step Solution

Answer

The two additional polar representations for the point (-3, -1.57) are (-3, 4.71) and (3, 1.57)

Step 1: Understand the Given Coordinate

The given polar coordinate is (-3, -1.57). In polar coordinates, the first value represents the distance from the origin 'r', and the second value represents the angle θ in radians. Here, r is -3 and θ is -1.57, which is approximately -π/2. Note that a negative radius means that the point is in the opposite direction of what θ suggests.

Step 2: Find Additional Representations Keeping Radius Same

Keeping the radius the same, we add 2π to the θ to get the another representation of the same point. So, (-3, -1.57 + 2π) = (-3, 4.71) is the another polar representation of the same point.

Step 3: Find Additional Representations Changing Sign of Radius

We can also change the sign of r and add or subtract π to θ to get another representation. So, changing sign of r and adding π, we get (3, -1.57 + π) = (3, 1.57) as another polar representation.

Key Concepts

Radians

Radians are a unit of angular measure. In polar coordinates, angles are commonly expressed in radians, which makes it easier to work with mathematical calculations, as this unit correlates directly with the properties of circles. One full revolution around a circle measures \( 2\pi \) radians, which is equivalent to 360 degrees. Thus, radians provide a natural way to represent angles, as they are closely linked to the dimensions of the circle itself.

In the given polar coordinate \((-3, -1.57)\), the angle \(-1.57\) is expressed in radians. This angle is roughly equal to \(-\frac{\pi}{2}\), indicating that the direction is 90 degrees in the negative angle direction. Remember that negative angles rotate clockwise while positive angles rotate counterclockwise. This understanding is crucial when plotting points in polar coordinates.

Angle Conversion

Angle conversion between radians and degrees is an essential skill, often necessary when working with polar coordinates or trigonometric functions. The conversion formula between degrees and radians is:

• To convert degrees to radians: \( ext{Radians} = ext{Degrees} \times \frac{\pi}{180} \)
• To convert radians to degrees: \( ext{Degrees} = ext{Radians} \times \frac{180}{\pi} \)

Using this conversion:

• Convert \(-1.57\) radians to degrees: \(-1.57 \times \frac{180}{\pi} \approx -90\) degrees.
• Therefore, a positive representation like \(4.71\) radians (from the solution) converts to \(4.71 \times \frac{180}{\pi} \approx 270\) degrees.

Converting angles can help you understand and visualize the position of points better on the polar grid. It allows for easier communication of the angles' concepts if you are more comfortable or familiar with degrees.

Negative Radius

The concept of a negative radius in polar coordinates can be initially perplexing. In standard practice, the radius \(r\) is a non-negative value representing the distance from the origin. When \(r\) is negative, it typically means the point is located in the direction opposite to the angle \(\theta\).In the coordinate \((-3, -1.57)\), the radius is negative, suggesting that the point is actually positioned 3 units in the direction diametrically opposed to \(-1.57\) radians.

To convert to an equivalent polar coordinate without changing location on the polar graph:

• We can adjust the angle by adding \(\pi\) (or 180°), while changing the radius to a positive value.
• For instance, \((-3, -1.57)\) converts to \( (3, 1.57) \) by making \( -1.57 + \pi = 1.57 \).

Understanding negative radius in polar coordinates helps visualize alternate equivalent points and gives more flexibility in representing points across different quadrants of the polar plane.