Problem 25 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

\(\theta = \arctan\left(-\frac{1}{2}\right)\)

Step 1: Understand the Cartesian Equation

The given Cartesian equation is \(2y = -x\). This can be rewritten as \(y = -\frac{x}{2}\). We'll use this form to help with the conversion to polar coordinates.

Step 2: Recall Polar-Cartesian Conversion Formulas

In polar coordinates, \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). We'll substitute these expressions into the Cartesian equation to convert it to polar form.

Step 3: Substitute Polar Expressions into the Equation

Substituting \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) into \(y = -\frac{x}{2}\), we get:\[ r \sin(\theta) = -\frac{1}{2} r \cos(\theta) \]

Step 4: Simplify the Polar Equation

We can simplify the equation \(r \sin(\theta) = -\frac{1}{2} r \cos(\theta)\) by dividing both sides by \(r\) (assuming \(r eq 0\)): \[ \sin(\theta) = -\frac{1}{2} \cos(\theta) \]

Step 5: Solve for \(\theta\)

Divide both sides of \(\sin(\theta) = -\frac{1}{2} \cos(\theta)\) by \(\cos(\theta)\) to obtain:\[ \tan(\theta) = -\frac{1}{2} \] This implies \(\theta = \arctan\left(-\frac{1}{2}\right)\).

Step 6: Conclude with Polar Form

Since the equation for \(\theta\) is constant, the polar equation is of the form:\[ \theta = \arctan\left(-\frac{1}{2}\right) \] This equation represents a line at a particular angle in polar coordinates.

Key Concepts

Conversion between Cartesian and Polar Coordinates

When dealing with mathematical graphs, it's important to know how to shift between different coordinate systems. Cartesian coordinates are the familiar \(x, y\) format, but sometimes polar coordinates, which use \(r\) and \(\theta\), can be more useful. Here's how you can convert from one to the other.To go from Cartesian to polar coordinates:

• The angle \(\theta\) is found using the arctangent function: \( heta = \arctan\left(\frac{y}{x}\right)\).
• The distance from the origin, \(r\), is calculated using Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\).

When converting the equation \(2y = -x\) into polar coordinates, you substitute the expressions for \(x\) (i.e., \(r \cos(\theta)\)) and \(y\) (i.e., \(r \sin(\theta)\)) into the original equation. After simplification, you find angles or specific orientation in the polar plane. This helps create equations that might be easier to interpret or solve within certain contexts.

Polar Equations

Polar equations differ from Cartesian equations in one key aspect: they focus on the relationship between the radial distance \(r\) and the angle \(\theta\). Rather than expressing \(y\) in terms of \(x\), polar equations revolve around angles and radii.An example of a polar equation is simply \(\theta = \text{constant}\), which represents a line at that particular angle from the origin. In the given exercise, this resulted in the polar equation \(\theta = \arctan\left(-\frac{1}{2}\right)\). This specifies a particular direction for the line, showing how polar coordinates give a unique perspective.Polar equations can represent a wide variety of shapes and graphs, from simple circles (e.g., \(r = \text{constant}\)) to more complex spirals and rose patterns depending on how \(r\) changes with \(\theta\). This versatility can make polar equations particularly powerful for understanding periodic processes and cyclical phenomena.

Trigonometric Functions

Trigonometric functions play a crucial role in polar coordinates. They provide the link between angle measurement and linear dimensions, making movement between Cartesian and polar forms possible.In polar coordinates:

• \(\sin(\theta)\) is used to express the vertical component of a point, equivalent to \(y\) in Cartesian coordinates.
• \(\cos(\theta)\) is used for the horizontal component, similar to \(x\) in Cartesian coordinates.
• The tangent function, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), is often used to determine the angle \(\theta\).

For instance, in converting the equation \(2y = -x\) to polar form, we utilized the relationship \(\sin(\theta) = -\frac{1}{2} \cos(\theta)\), ultimately simplifying this to \(\tan(\theta) = -\frac{1}{2}\), a function defined strictly by angle.Trigonometric functions underpin most of polar coordinate's power by allowing the transformation of coordinate expressions in flexible, dynamic ways.