Problem 52 - 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

(a) Graph resembles a parabola; (b) Rectangular form is \( x^2 = 9(y - 1) \).

Step 1: Graph the Polar Equation

To graph the polar equation \( r=\tan \theta \sec \theta \) in the specified viewing rectangle, note that \( r = 1/\cos \theta \) because \( \sec \theta = 1/\cos \theta \). Therefore, \( r = \tan \theta / \cos \theta = \sin \theta / \cos^2 \theta \). Graph this equation within \( \theta \) bounds such that \( \theta \) causes \( \cos \theta eq 0 \) to avoid division by zero. For \( \theta \) from \( -\pi/2 \) to \( \pi/2 \), the output resembles the form of a parabola.

Step 2: Convert to Rectangular Coordinates

To confirm the parabola shape, convert to rectangular coordinates. Recall the conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \). From the polar equation \( r = \tan \theta \sec \theta = \frac{\sin \theta}{\cos^2 \theta} \), substitute \( \tan \theta = \sin \theta/\cos \theta \) and \( \sec \theta = 1/\cos \theta \). Thus, \( y = r \sin \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \). Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), rewrite as \( x = \frac{y}{\cos \theta} \). By squaring both sides and simplifying, the equation resembles \( x^2 = 9(y - 1) \), which is the equation of a parabola.

Key Concepts

Graphing Polar Equations

Graphing polar equations can be a fun and interesting way to visualize curves. Polar equations are different from regular (or Cartesian) equations because they use the radius and angle for plotting. In the exercise, we work with the polar equation \( r = \tan \theta \sec \theta \).
To graph polar equations effectively, you adhere to the provided viewing window. For our case, it is \([-3,3]\) by \([-1,9]\). A polar equation uses the angle \( \theta \) and radius \( r \). This equation is transformed using trigonometric identities to make it easier to plot. Remember that \( \sec \theta = 1/\cos \theta \), so our equation becomes \( r = \sin \theta / \cos^2 \theta \).

• Keep in mind that polar graphs can have exotic shapes. Use tools like graphing software to visualize your result.
• Focus on the behavior of the equation when \( \cos \theta = 0 \). This helps avoid dividing by zero in your graph.
• For this equation, observe it from \(-\pi/2\) to \(\pi/2\) to ensure a complete graph.

In this range, the output of the graph should resemble a parabola.

Rectangular Coordinates Conversion

Converting polar coordinates to rectangular coordinates allows us to work with more familiar graph layouts. In this problem, the given polar equation hints at a parabola shape.
This conversion follows basic steps:
1. Use the identity \( x = r \cos \theta \) and \( y = r \sin \theta \).
2. Substituting them from the polar form gives \( r = \tan \theta \sec \theta = \sin \theta / \cos^2 \theta \).

• We solve for \( y \) in terms of \( \sin \theta \) by multiplying \( r \) by \( \sin \theta \), resulting in \( y = \sin^2 \theta / \cos^2 \theta \).
• Next, leverage the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to frame the answer.
• Connect the dots back to rectangular form by making \( x = frac{y}{\cos \theta} \) and manipulating it, finding \( x^2 = 9(y - 1) \).

Thus, the polar form represents a parabola when converted.

Parabola in Polar Coordinates

Seeing a parabola in polar coordinates might seem unique, but it's a marvelous example of discovering hidden geometries. Polar equations occupied by trigonometric functions can shape into familiar forms like parabolas when viewed correctly.
This exercise exhibits this by showing that \( r = \tan \theta \sec \theta \) converts to a familiar parabola equation when interpreted into rectangular coordinates. This highlights the interconnectedness between different systems of graphing, mainly polar and Cartesian.

• Such transformations enhance our understanding of different mathematical representations.
• Viewing a polar graph as a parabola opens up new perspectives on how polar equations express conic sections.
• Learn to appreciate the variance in coordinate systems, as every form provides new insights and solutions.

Understanding these relationships can deepen your comprehension of geometry and the ability to predict graph shapes from equations.