Problem 9 - College Trigonometry

Question

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ (-20,3 \pi) $$

Step-by-Step Solution

Answer

(a) \((-20, \pi)\), (b) \((20, 0)\), (c) \((20, 5\pi)\)

Step 1: Convert Polar Point

The given polar point is \((-20, 3\pi)\). In polar coordinates, the radius \(r\) and the angle \(\theta\) are given. Here, \(r = -20\) and \(\theta = 3\pi\).

Step 2: Express (a): Convert to r < 0, θ in [0, 2π]

The point \((-20, 3\pi)\) already satisfies the condition \(r < 0\). However, the angle must be adjusted as \(3\pi\) is not between 0 and \(2\pi\). Use the identity \((-20, 3\pi) \equiv (-20, 3\pi - 2\pi) = (-20, \pi)\). So, the point is \((-20, \pi)\).

Step 3: Express (b): Convert to r > 0, θ ≤ 0

To meet the condition \(r > 0\), we change the sign of \(r\) and adjust \(\theta\) by subtracting \(\pi\) to keep the point the same in Cartesian coordinates: \((-20, 3\pi) \equiv (20, 3\pi - \pi) = (20, 2\pi)\). Since \(2\pi\) is not less than or equal to 0, add another full circle to make it negative: \((20, 2\pi - 2\pi) = (20, 0)\).

Step 4: Express (c): Convert to r > 0, θ ≥ 2π

For positive \(r\) and \(\theta \geq 2\pi\), retain the positive \(r\) and add \(2\pi\) to the angle \((20, 3\pi)\): \((20, 3\pi) \equiv (20, 3\pi + 2\pi) = (20, 5\pi)\). This ensures \(\theta\) is greater than \(2\pi\).

Key Concepts

Negative Radius in Polar Coordinates

In polar coordinates, a point is defined by a pair \(r, \theta\), where \r\ is the radius or distance from the origin, and \theta\ is the angle with the positive x-axis.
Unlike Cartesian coordinates, polar coordinates can have a negative radius. When \r\ is negative, the point is located in the opposite direction on the radial line determined by \theta\.
For example, if a point is given as \(-20, 3\pi\), the negative radius \-20\ indicates that we must look opposite in direction along the line formed by the angle \3\pi\.
This opposite direction flips the location of the point along this radial line, ensuring the orientation is maintained but essentially pointing to the opposite of where you would end if \r\ was positive.

• To adjust \(\theta\), calculations such as \(\theta - 2\pi\) are used to ensure it falls within a standard interval.

Understanding negative radii helps visualize how polar coordinates can represent points with flexibility, accommodating convenience in calculation and graphic representation.

Angle Transformation in Polar Coordinates

Transforming angles in polar coordinates can seem tricky, yet it provides powerful insights into movement around the circle.
The transformation primarily deals with keeping the angle \theta\ within desirable boundaries, like \[0, 2\pi)\] for convenience.
In the exercise provided, the point \(-20, 3\pi\) has an angle of \3\pi\, which surpasses the typical \[0, 2\pi)\] range.
To transform this angle to fit \[0, 2\pi)\], you simply subtract \2\pi\, resulting in an equivalent representative angle.

• For cases such as negative \(\theta\), move backward by adding multiples of \2\pi\. This effectively twists the polar plot clock or counterclockwise as needed.

Thinking of polar angles as ascertainable multiples of a full circle helps to conceptualize how movement is expressed around the origin in polar systems.

Conversion of Polar to Cartesian Coordinates

The conversion from polar to Cartesian coordinates is a common requirement when analyzing geometrical points.
In Cartesian coordinates, points are represented as \(x, y\), located directly on the x and y grid, whereas polar coordinates feature radial and angular designations. To convert a point from polar to Cartesian coordinates:

• Use the formula \(x = r \cdot \cos(\theta)\).
• Calculate \(y = r \cdot \sin(\theta)\).

For example, if you have a point \(20, 5\pi\), with r positive and beyond the general \[0, 2\pi)\] angle range, plugging these into the formulas yields precise Cartesian coordinates illustrating the same point.
This conversion facilitates interaction between different graphing modes, enabling seamless transitions between problems demanding polar and Cartesian interpretations.