Problem 20 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}=8 y$$

Step-by-Step Solution

Answer

The polar equation is \(r = 8 \tan(\theta) \sec(\theta)\).

Step 1: Understand the Cartesian Equation

The given equation is \(x^2 = 8y\). This is a parabola that opens upwards. This equation is in Cartesian form, where \(x\) and \(y\) are coordinates on a plane.

Step 2: Convert Cartesian to Polar Coordinates

In polar coordinates, \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Substitute these into the Cartesian equation to begin the conversion.

Step 3: Substitute Polar Equations

Replace \(x\) with \(r \cos(\theta)\) and \(y\) with \(r \sin(\theta)\) in the equation \(x^2 = 8y\). Thus, it becomes \((r \cos(\theta))^2 = 8(r \sin(\theta))\).

Step 4: Simplify the Polar Equation

Simplify the equation from Step 3: \(r^2 \cos^2(\theta) = 8r \sin(\theta)\). Divide both sides by \(r\) (assuming \(r eq 0\)) to get \(r \cos^2(\theta) = 8 \sin(\theta)\).

Step 5: Rearrange to Find Polar Equation

The polar equation becomes \(r = \frac{8 \sin(\theta)}{\cos^2(\theta)}\), which is equivalent to \(r = 8 \tan(\theta) \sec(\theta)\) using trigonometric identities.

Key Concepts

Cartesian coordinates

Cartesian coordinates are a fundamental system used for graphing points on a plane. They consist of two axes, usually labeled as the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is represented by an ordered pair (x, y), where:

• x is the distance from the y-axis, showing horizontal movement.
• y is the distance from the x-axis, showing vertical movement.

The Cartesian coordinate system is particularly useful because it allows us to represent geometric shapes and equations conveniently. For example, a simple equation like \(x^2 = 8y\) can describe the shape of a parabola on this grid. This format allows easy visualization and manipulation of geometric figures, making it a powerful tool in mathematics.

parabola

A parabola is a symmetrical, open curve that represents a particular set of quadratic equations, often noted in their simplest form as \(y = ax^2 + bx + c\). However, they can appear in many orientations. In the equation \(x^2 = 8y\), the parabola opens upwards with its vertex at the origin. Some key properties of parabolas include:

• The vertex is the parabola's peak or lowest point, depending on its orientation.
• The axis of symmetry is a line through the vertex, dividing the parabola into two mirror-image halves.
• The focus and directrix help define the parabola, influencing its width and direction.

Parabolas are naturally found in scenarios involving projectile motions, satellite dishes, and even the paths of bouncing balls. Understanding them in Cartesian coordinates is key to comprehending their behaviors and properties.

coordinate conversion

Coordinate conversion involves transforming a set of coordinates in one system to another, such as moving from Cartesian to polar coordinates. This process is key in various fields like physics, engineering, and navigation where different coordinate systems simplify different problems.In the case of converting the equation \(x^2 = 8y\) into polar coordinates:

• Use the equations \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to substitute for x and y.
• The substitution changes the equation into one that involves r and \(\theta\), the polar coordinates.

This step makes it easier to handle problems involving angles and radii, especially when dealing with circular and rotational systems. Polar coordinates often simplify the mathematics of systems with symmetry or radial dependencies.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are essential tools in simplifying complex expressions and solving equations, especially when converting between coordinate systems.In converting \(x^2 = 8y\) from Cartesian to polar coordinates, two important identities are utilized:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

These identities help transform the equation into a form that uses trigonometric functions, making them immensely useful in various applications, including physics and engineering. Understanding trigonometric identities can reveal simplifications that aren't apparent otherwise, helping to provide clear and elegant solutions to mathematical problems.