Problem 14 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=-3 \sec \theta$$

Step-by-Step Solution

Answer

The polar equation \(r = -3 \sec \theta\) corresponds to a half-line on the left side of the Cartesian plane with x = -3, excluding the points where \(y = (2n+1) \pi\).

Step 1: Understanding \(-3 \sec \theta\)

The secant function, \(\sec \theta\), is the reciprocal of the cosine function, i.e \(\sec \theta = 1/ \cos \theta\). Therefore, \(r = - 3 \sec \theta\) simplifies to \(r = -3 / \cos \theta\).

Step 2: Transform to rectangular coordinates

In polar coordinates, \(r = -3 / \cos \theta\). Thus, to transform this into rectangular coordinates, we'll make use of the relation \(r \cos \theta = x\), giving us \(x = -3\).

Step 3: Sketch the graph

The equation \(x=-3\) presents a vertical line that crosses the x-axis at -3. However, because values of \(r\) can never be negative in polar coordinates, the line \(x=-3\) does not change across the y-axis, giving a half-line starting from the 'pole' (origin) and goes to the left.

Step 4: Domain Consideration

The cosine function is not defined for certain angles, causing the secant function to also be undefined. For \(\cos \theta\), where \(\theta = (2n+1) \pi /2\), \(\cos \theta = 0\). Thus secant \(\theta\) is undefined at these points. This will result in breaks in the line \(x=-3\) at all points where \(y = (2n+1) \pi\). This means the plot of the polar equation \(r= -3\sec \theta\) is a half-line \(x=-3\) excluding these points.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane through a distance from a reference point (usually the origin) and an angle from a reference direction (typically the positive x-axis). This system is particularly useful in situations where direction and distance from a central point are more intuitive than traditional x and y coordinates. In our exercise, we deal with the equation \(r = -3 \sec \theta\), where \(r\) is the radial distance and \(\theta\) is the angular direction.

• The variable \(r\) denotes the radius and connects to how far away you are from the origin.
• The angle \(\theta\) defines direction, measured counterclockwise from the positive x-direction.
• Remember, negative values for \(r\) can suggest movement in the opposite direction of the angle \(\theta\).

This equation portrays a vertical line, despite the polar nature, due to how these coordinates convert into rectangular form.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, such as secant (\(\sec \theta\)), are derived from the primary trig functions by way of reciprocals. This means you're dividing one by the original trigonometric function.

• The secant function is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
• In our problem, \(r = -3 \sec \theta\), translates to \(r = -\frac{3}{\cos \theta}\).

These functions unveil new insights into the behavior of trigonometric graphs and equations, especially when they bridge between different coordinate systems as seen here.

Rectangular Coordinates

Rectangular coordinates represent points in the conventional \((x, y)\) format, beneficial for many common applications and graphical interpretations. Translating from polar to rectangular coordinates aids in visualizing polar equations within a familiar framework.

• With \( r = -3 \sec \theta\), the transformation uses the relationship \(r \cos \theta = x\).
• Substituting gives \(x = -3\), representing a vertical line at \(x = -3\).

This transition underscores the way polar equations manifest into straightforward linear representations in the rectangular plane, allowing for easy visualization and graph sketching.

Domain Considerations

In mathematics, understanding where a function is defined is critical to accurately plotting graphs and ensuring logical integrity. With reciprocal trigonometric functions, like our \(\sec \theta\), some angles make these functions undefined.

• The secant function becomes undefined whenever \(\cos \theta = 0\).
• This occurs at \(\theta = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer.

In the specific equation \(r=-3 \sec \theta\), these points lead to interruptions in the graph, as you cannot plot points where the function misbehaves (undefined values). Thus, you get gaps in the plotted vertical line at \(x = -3\). Understanding these nuances ensures accurate and complete graph plotting.