Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ 2 y=-x $$

Step-by-Step Solution

Answer

The polar equation is \(r = 0\) or \(\theta = \arctan\left(-\frac{1}{2}\right) + k\pi\).

Step 1: Convert Cartesian Equation to Polar Form

The given equation is in Cartesian coordinates: \(2y = -x\). Recall the conversion formulas: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substitute these into the equation: \(2(r\sin(\theta)) = -(r\cos(\theta))\).

Step 2: Simplify Polar Equation

Factor the \(r\) from both sides: \(r(2\sin(\theta) + \cos(\theta)) = 0\). This can be split into two separate equations: either \(r = 0\) or \(2\sin(\theta) + \cos(\theta) = 0\). Here, \(r = 0\) corresponds to the pole, and it is part of the solution.

Step 3: Resolve\(2\sin(\theta) + \cos(\theta) = 0\)

Focus on solving \(2\sin(\theta) + \cos(\theta) = 0\). Rearrange to get \(2\sin(\theta) = -\cos(\theta)\). Divide both sides by \(\cos(\theta)\) to obtain \(2\tan(\theta) = -1\). This simplifies to \(\tan(\theta) = -\frac{1}{2}\), giving the specific angles \(\theta = \arctan\left(-\frac{1}{2}\right)\ + k\pi\) (where \(k\) is an integer).

Step 4: Formulate Complete Polar Equation

Consider the implications of the solution. \(r = 0\) implies certain points (usually radial lines through the origin), and \(\theta = \arctan\left(-\frac{1}{2}\right)\ + k\pi\) describes the valid angular directions of a line. Thus, the polar equation equivalent is \(r = 0\) or \(\theta = \arctan\left(-\frac{1}{2}\right) + k\pi\), demonstrating the same linear relationship: \(r = \frac{-\cos(\theta)}{2\sin(\theta)} = \tan^{-1}(-\frac{1}{2}) + k\pi\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a way to pinpoint a location on a plane by using two numbers, usually called x and y. Imagine a flat surface with two crossing lines — one horizontal (the x-axis) and one vertical (the y-axis). The intersection point is called the origin, marked by (0, 0) and it's the starting point for locating other points.

• The x-coordinate tells you how far to move along the horizontal path.
• The y-coordinate indicates the distance to move along the vertical path.

When given in the form of an equation, like the one you've seen (2y = -x), it describes a relationship between x and y, determining a straight line on this plane. On a graph, every pair of x and y satisfying this equation lies on this line. This method is hugely valuable in many mathematical applications, such as geometry and algebra.

Polar Coordinates

Polar coordinates use a different approach from the Cartesian system to describe a point in a plane. Instead of utilizing two distances, it uses a distance and an angle. Think about standing at the origin and looking at a particular point:

• The distance from you to the point is called r (radius).
• The angle between a fixed direction, typically the positive x-axis, and a line connecting you to the point is the angle \(\theta\).

Using (r, \(\theta\)), polar coordinates allow you to describe any point on a plane with a unique perspective. This representation is particularly useful for dealing with curves or systems where angles and distances are of primary concern, such as circular patterns or periodic functions.

Trigonometric Identities

Trigonometric identities are crucial tools that relate angles and sides in right-angled triangles. They also apply to convert between different forms of equations, particularly in transformations like converting Cartesian equations to polar form. Some key identities include:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

These identities not only simplify complex trigonometric expressions but also assist in identifying relationships when switching coordinate systems. For instance, by using \(\tan(\theta)\), you can easily relate an angle back to the x and y relationships in Cartesian form.

Conversion Formulas

Conversion formulas bridge the gap between different coordinate systems. They are like a translator between Cartesian coordinates and polar coordinates. By understanding and using these formulas, you can express any point in one system as equivalent in the other. The main conversion, as used in your exercise, involves the following:

• Cartesian to Polar: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\)
• Polar to Cartesian: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\)

These conversion techniques are essential when you need to graph equations more naturally presented in one form than the other. They also allow seamless shifting between perspectives, allowing one to choose the most appropriate system depending on the problem at hand. It's a vital step in enhancing the understanding and application of mathematical concepts across different contexts.