Problem 58 - Algebra and Trigonometry

Question

(a) Graph the polar equation \(r=\tan \theta \sec \theta\) in the viewing rectangle \([-3,3]\) by \([-1,9]\) . (b) Note that your graph in part (a) looks like a parabola (see Section \(3.5 ) .\) Confirm this by converting the equation to rectangular coordinates.

Step-by-Step Solution

Answer

Graph the polar equation and confirm it represents the parabola \(y = x^2\).

Step 1: Understanding the Polar Equation

The given polar equation is \(r = \tan \theta \sec \theta\). To graph this, we need to plot points where \(\theta\) is within the standard polar coordinate limits, typically \(0\) to \(2\pi\), and calculate corresponding \(r\) values.

Step 2: Graph the Polar Equation

Using the polar equation \(r = \tan \theta \sec \theta\), compute several points across the desired range. The values of \(\theta\) should be chosen such that \(\tan \theta\) and \(\sec \theta\) are defined. Once calculated, plot these points within the viewing rectangle \([-3,3]\) by \([-1,9]\). The graph should resemble a parabola.

Step 3: Convert to Rectangular Coordinates

To convert \(r = \tan \theta \sec \theta\) into rectangular coordinates, we utilize the polar-to-rectangular conversions: \(x = r \cos \theta\) and \(y = r \sin \theta\). First, rewrite \(r\cos\theta = \tan\theta\cos\theta\). Since \(x = r\cos\theta\), we have \(x = \sin\theta\). Also, \(y = r\sin\theta = \sin\theta\sec\theta\tan\theta\). Since \(\sec\theta = \frac{1}{\cos\theta}\), the equation simplifies to \(y = \sin^2\theta/\cos\theta\). Recall, \(\tan\theta = \sin\theta/\cos\theta\), so \(y = \tan^2\theta\).

Step 4: Simplify the Rectangular Equation

Replace \(\tan\theta = x\) (since \(x=\sin\theta\)), so \(y = x^2\). This result is the equation of a parabola in rectangular coordinates.

Key Concepts

Rectangular Coordinates

In the world of mathematics, coordinates are essential for mapping points on a plane. Rectangular coordinates, also known as Cartesian coordinates, are a fundamental system for doing just that. They use two axes—the horizontal axis (x) and the vertical axis (y)—to define a point's position. Every point is expressed as an ordered pair (x, y). Each pair uniquely identifies a location on the coordinate grid.

Rectangular coordinates are particularly useful because they form the basis for a wide range of mathematical functions and equations. When it comes to graphing, rectangular coordinates simplify equations and allow for easy visualization. For instance, lines, curves, and shapes like parabolas can be represented cleanly using this system. Linear functions or parabolas follow distinct patterns on this grid, making them easy to analyze and interpret.

Polar to Rectangular Conversion

Converting between polar and rectangular coordinates can sometimes seem complex, but it's merely a way to express the same point in different systems. Polar coordinates use distance from a reference point (radius, r) and angle from a reference direction (theta, \(\theta\)) to pinpoint locations on a plane.

To switch from polar to rectangular, specific formulas are used:

• x = r \(\cos \theta\)
• y = r \(\sin \theta\)

These conversion equations translate the radial and angular components into the familiar x and y coordinates.

For example, if you have a polar equation like \( r = \tan \theta \sec \theta \), you can convert it to rectangular form using these equations. This allows mathematicians to represent and manipulate equations across different systems, simplifying analysis and graphing.

Parabola Graphing

Graphing a parabola is an intriguing process that makes use of both geometry and algebra. In rectangular coordinates, the standard equation of a parabola is usually shown as \( y = ax^2 + bx + c \). Parabolas have a distinct U-shape or inverted U-shape, and their orientation, size, and position are influenced by the coefficients a, b, and c.

When analyzing the graph of a polar equation that resembles a parabola, conversion to rectangular coordinates often helps confirm its parabolic nature. Once the equation is expressed in the familiar \( y = x^2 \) form, plotting it becomes straightforward. Checking values of x and computing corresponding y-values will give the curve's distinct shape.

Recognizing these parabolic shapes within different coordinate systems is crucial, as it allows for better interpretation and application to real-world scenarios.

Trigonometric Identities

Trigonometric identities are crucial tools in mathematics, particularly for simplifying complex expressions and solving equations. They involve relationships among the functions sine, cosine, tangent, and others that connect angles and sides of triangles. Mastering these identities is key when working with polar coordinates and transformations.

Some of the basic and commonly used trigonometric identities include:

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

Using these identities, you can simplify terms during polar to rectangular conversion and understand their relationships better. In the equation \( r = \tan \theta \sec \theta \), these identities help in expressing equations more conveniently, supporting better understanding and graphing.