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Chapter 10: Problem 64

Sketch the graph of the polar equation. $$r=-3 \sec \theta$$

Short Answer

Expert verified
Convert to \(x = -3\) in Cartesian coordinates and sketch it as a vertical line.

Step by step solution

01

Understanding the Trigonometric Relationship

The equation given is in polar form: \(r = -3 \sec \theta\). The secant function, \(\sec \theta\), is the reciprocal of the cosine function. Therefore, we can rewrite the equation using cosine: \(r = -3/\cos \theta\).
02

Identifying Cartesian Equivalence

To graph a polar equation, it's often useful to convert it to Cartesian coordinates (\(x, y\)). Recall the relationships: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r = \sqrt{x^2 + y^2}\). Substitute these into our equation \(r = -3/\cos \theta\). Simplifying gives \(x = -3\).
03

Sketching the Graph

Based on the conversion, \(x = -3\), this indicates a vertical line in the Cartesian plane where \(x\) is always \(-3\) and \(y\) can be any value. In polar coordinates, this represents a circle centered at the origin with radius \(3\), but flipped horizontally across the origin due to the negative sign.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Equations
Polar equations are a unique way to describe curves using the polar coordinate system. Instead of relying on the traditional Cartesian setup of x and y, polar coordinates involve a radius and an angle to define a point.
In our exercise, the polar equation given is \(r = -3 \sec \theta\). This represents the relationship between the radius \(r\) and the angle \(\theta\).
  • The radius \(r\) can be positive or negative, with negative values indicating a point in the opposite direction.
  • The angle \(\theta\) measures in radians, typically between 0 and \(2\pi\).
  • The secant function, \(\sec \theta\), is involved, which connects to trigonometric functions.
Understanding the polar equation provides vital insights into the curve's shape and orientation. In this case, transforming \(r = -3 \sec \theta\) helps simplify the sketching process by finding the Cartesian equivalent.
Trigonometric Functions
Trigonometric functions like sine, cosine, and secant play a pivotal role in both polar and Cartesian forms. In our scenario, the secant function \(\sec \theta\) is crucial for expressing our polar equation.
Remember:
  • \(\sec \theta\) is the reciprocal of \(\cos \theta\).
  • Being able to convert secant to cosine simplifies handling the equation.
  • Trigonometric functions describe angles and can reveal patterns in graphs.
When dealing with \(r = -3 \sec \theta\), converting it to \(r = \frac{-3}{\cos \theta}\) clarifies how radii vary with angle changes. Recognizing this relationship aids in shifting from polar to Cartesian coordinates, making graphing simpler.
Cartesian Coordinates
Cartesian coordinates allow a more familiar representation method, using x and y to plot points. Converting polar equations to this system can simplify the graphing task.
In the exercise, the polar equation \(r = -3 \sec \theta\) transitions to Cartesian coordinates:
  • Use \(x = r \cos \theta\)
  • Use \(y = r \sin \theta\)
  • \(r\) converts based on these trigonometric identities.
Plugging \(r = \frac{-3}{\cos \theta}\) into \(x = r \cos \theta\) gives \(x = -3\). This describes a vertical line where \(x = -3\), independent of \(y\). In polar terms, this line illustrates a circle centered at the origin with radius 3, showcasing symmetry and transformations unique to polar-to-Cartesian conversions.

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