Problem 71 - Precalculus with Limits

Question

Sketch the graph of \(r\ =\ 6\ \cos\ \theta\) over each interval.Describe the part of the graph obtained in each case. (a) \(0 \leq\ \theta\ \leq\ \dfrac{\pi}{2} \quad \quad\) (b) \(\dfrac{\pi}{2} \leq\ \theta\ \leq\ \pi\) (c) \(-\dfrac{\pi}{2} \leq\ \theta\ \dfrac{\pi}{2}\ \quad \quad\) (d) \(\dfrac{\pi}{4} \leq\ \theta\ \leq\ \dfrac{3\pi}{4}\)

Step-by-Step Solution

Answer

The graph for \(r = 6\cos\theta\) will be a circle centered on the positive x-axis with a radius of 3. For each interval of \(\theta\), it will cover different parts of the circle. The parts of the circle covered are: (a) first quadrant; (b) second quadrant; (c) half-circle in the first and fourth quadrants; (d) an eighth of the circle in the first and second quadrant.

Step 1: Identify the Curve

\(r = 6\cos\theta\) is a circle with center \((3, 0)\) and radius 3 in Cartesian coordinates.

Step 2: Sketch Over Each Interval

(a) \(0 \leq \theta \leq \frac{\pi}{2}\): The upper semicircle (right half, \(r \geq 0\)).
(b) \(\frac{\pi}{2} \leq \theta \leq \pi\): \(r\) is negative in this interval, so the curve traces the lower semicircle.
(c) \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\): The full circle is traced.
(d) \(\frac{\pi}{4} \leq \theta \leq \frac{3\pi}{4}\): This traces the upper half of the circle.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to specify points in a plane by two values: the radial coordinate "\(r\)" and the angular coordinate "\(\theta\)". Unlike the Cartesian system, which uses x and y coordinates to indicate position, polar coordinates focus on the distance from the origin and the angle from a reference direction, typically the positive x-axis.

Here is how you can think about it:

• Radial Coordinate \(r\): It represents the distance from the origin to the point. Think of it like the radius of a circle.
• Angular Coordinate \(\theta\): It indicates the direction in which the point is located from the origin. The angle is measured in radians or degrees from the positive x-axis.

Polar coordinates are incredibly useful in contexts where the relationship between points is naturally circular, such as when dealing with rotations or periodic motion. They can express complex shapes quite simply, such as the circle represented by the equation \(r = 6\cos\theta\) in the given exercise.

Graphing Polar Equations

Graphing polar equations can initially seem challenging, but it becomes much easier once you understand how polar coordinates work.

• The equation \(r = 6\cos\theta\) describes a circle because \(r\) changes depending on the cosine of the angle \(\theta\), with a maximum value of 6 when \(\theta = 0\).
• The nature of the trigonometric function \(\cos\theta\) dictates that maximum positive \(r\) values occur between \(0\) and \(\pi/2\), which graphically means the circle is positioned on the positive x-axis.
• Each defined interval of \(\theta\) determines a specified portion of the circle to sketch, such as quarter-circles within certain quadrants.

When graphing these equations, begin by determining the range of \(\theta\) values for which you will sketch the graph. Consider the trigonometric function values at key angles such as \(0\), \(\pi/4\), \(\pi/2\), etc. Visualizing these values helps in sketching the segment of the circle corresponding to each part of \(\theta\). Finding symmetry can also simplify graphing, especially when \(\theta\)'s interval spans multiple quadrants.

Trigonometric Functions

Trigonometric functions, like cosine, play a crucial role in expressing polar equations. With polar coordinates, the trigonometric functions determine how far a point is from the pole based on its angle.

• The \(\cos\theta\) function specifically affects how the radius \(r\) varies with respect to the angle \(\theta\), making \(r = 6\cos\theta\) a dynamic relationship where \(r\) stretches or shrinks as \(\theta\) changes.
• The cosine function ranges from -1 to 1, creating a loop effect whereby the radius reduces to zero at \(\pi/2\) and then begins to "mirror" itself around the y-axis for negative values of cosine.
• Such behavior is why the equation \(r = 6\cos\theta\) graphically traces out a circle centered on the x-axis.

Understanding how trigonometric functions map angles to coordinates radically simplifies the process of sketching complex curves in polar coordinates. Recognizing the periodic nature of these functions and how they alter the radial distance is fundamental to mastering polar graphing.