Problem 16 - 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 \(\theta \geq 2 \pi\) $$ (-\pi,-\pi) $$

Step-by-Step Solution

Answer

Expressions: (-π, π), (π, 0), (π, 3π).

Step 1: Understanding Polar Coordinates

Polar coordinates are expressed as \(r, \theta\), where \(r\) represents the radius (distance from the origin) and \(\theta\) represents the angle from the positive x-axis. In the point \((-\pi, -\pi)\), \(r = -\pi\) and \(\theta = -\pi\) radians.

Step 2: Convert to Cartesian Coordinates

To better understand polar coordinates, convert \((-\pi, -\pi)\) into Cartesian coordinates using \(x = r \cos \theta\) and \(y = r \sin \theta\). Here, \(x = (-\pi) \cdot \cos(-\pi) = \pi\) and \(y = (-\pi) \cdot \sin(-\pi) = 0\). Thus, in Cartesian coordinates, the point is \((\pi, 0)\).

Step 3: Expressing (a) \((r < 0, 0 \leq \theta \leq 2\pi)\)

We need to express the point with a negative radius and positive angle. Since \(\theta\) is \(-\pi\), add \(2\pi\) to it to shift it within the range: \(-\pi + 2\pi = \pi\). The point is represented as \((-\pi, \pi)\).

Step 4: Expressing (b) \((r > 0, \theta \leq 0)\)

For a positive radius and non-positive angle, invert the radius and adjust the angle. The radius becomes positive \((\pi)\), and we shift \(\theta\) by adding \(\pi\) (to reach zero). Thus, the point is \(\pi, 0 \).

Step 5: Expressing (c) \((r > 0, \theta \geq 2\pi)\)

Finally, for a positive radius and angle greater than or equal to \(2\pi\), add \(2\pi\) to our current \(\theta\). With \(r = \pi\) (positive) and \(\theta = -\pi + 4\pi = 3\pi\), the point is \(\pi, 3\pi\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a fundamental concept in mathematics, useful for plotting points on a two-dimensional plane. Here, any location is identified using an ordered pair \(x, y\), which represents the horizontal and vertical distances from an origin point. This system is particularly intuitive as it relates directly to our everyday understanding of space.

• x-coordinate: The horizontal distance from the origin. It tells how far left or right a point is from the vertical axis.
• y-coordinate: The vertical distance from the origin. It indicates how far up or down a point is from the horizontal axis.

In the context of converting polar coordinates to Cartesian coordinates, equations like \(x = r \cos \theta\) and \(y = r \sin \theta\) are extremely helpful. They allow us to map positions described in terms of radius and angle, into the more familiar Cartesian plane.

radius

In polar coordinates, the radius \(r\) is essential. It defines the distance from the origin to a specific point in the plane. This is not much different from the radius of a circle, which is the consistent distance from the center to any point on its boundary.

• A positive radius means that the point is in the intended direction from the origin.
• A negative radius flips the direction through the origin to the other side of the circle.

This flipping effect underscores an interesting aspect of polar coordinates—the same point can often be expressed in multiple ways! By adjusting the values of \(r\) and \(\theta\), you can represent the same location with negative radius values by repositioning the angle.

angle

In polar coordinates, the angle \(\theta\) is just as crucial as the radius. It determines the direction from the positive x-axis where the radius extends. Angles in polar coordinates are generally measured in radians, a natural choice in mathematics for measuring circular shapes.

• Angles help define the rotational placement of a point.
• Typically, angles in polar coordinates are represented starting from 0, going counterclockwise around the unit circle.

This concept enables a straightforward method for navigating through a circular path and forms the backbone for expressing polar coordinates. Adjusting \(\theta\) impacts where the point lies, reinforcing the idea that the same point can be represented with multiple angle values.

angle conversion

Converting angles in polar coordinates often involves shifting the angle \(\theta\) by full circular turns (usually represented as \(2\pi\) in radians). This conversion is key for re-expressing points.

• A subtraction or addition of \(2\pi\) moves the angle around by a full circle.
• When dealing with negative angles, adding \(2\pi\) helps to convert them into an equivalent positive angle.
• This flexibility allows points to be expressed uniquely based on constraints provided, such as ensuring \(0 \leq \theta \leq 2\pi\) or other specified ranges.

With angle conversion, you can smoothly navigate through different representations in polar coordinates, ensuring proper alignment with the given conditions of an exercise.