Problem 28 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r=\cos \theta-1 $$

Step-by-Step Solution

Answer

The graph is a cardioid centered at the pole, touching the origin.

Step 1: Interpret the Polar Equation

The given polar equation is \( r = \cos \theta - 1 \). This implies that the radius \( r \) depends on the angle \( \theta \), and it is calculated by taking the cosine of \( \theta \) and then subtracting 1.

Step 2: Determine Key Values of \( \theta \)

To understand the behavior of the polar graph, calculate \( r \) for specific values of \( \theta \). Start with important angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).

Step 3: Calculate \( r \) for Selected Angles

- For \( \theta = 0 \): \( r = \cos 0 - 1 = 1 - 1 = 0 \).- For \( \theta = \frac{\pi}{2} \): \( r = \cos \frac{\pi}{2} - 1 = 0 - 1 = -1 \).- For \( \theta = \pi \): \( r = \cos \pi - 1 = -1 - 1 = -2 \).- For \( \theta = \frac{3\pi}{2} \): \( r = \cos \frac{3\pi}{2} - 1 = 0 - 1 = -1 \).- For \( \theta = 2\pi \): \( r = \cos 2\pi - 1 = 1 - 1 = 0 \).

Step 4: Sketch the Graph

Plot the points found from the calculated \( r \) values on the polar coordinate system. Connect these points to visualize the graph. At \( \theta = 0 \), the graph begins at the origin (where \( r = 0 \)). At \( \theta = \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the graph goes to a radius of -1, meaning it crosses into the opposite side. At \( \theta = \pi \), the radius is at -2. As \( \theta \) approaches \( 2\pi \), the graph returns to the origin. The complete graph resembles a heart shape known as a "cardioid".

Key Concepts

Understanding Polar Coordinates

Polar coordinates provide a way to specify the location of a point in a plane. Instead of using the traditional x and y coordinates, polar coordinates use a distance from a reference point and an angle from a reference direction.

• The distance, known as the radius, is denoted by \( r \) and measures how far the point is from the pole (similar to the origin in Cartesian coordinates).
• The angle, denoted by \( \theta \), is measured from the positive x-axis (a reference direction) in a counter-clockwise direction.

To determine a point's polar coordinates:\( (r, \theta) \), you usually start at the origin, then rotate by \( \theta \) degrees or radians, and finally move outwards by \( r \) units.
Converting between Cartesian and polar coordinates can often be useful. Use the formulas: \( x = r \cos \theta \), \( y = r \sin \theta \).
Understanding polar coordinates is crucial for tasks involving circular or rotational systems.

The Art of Graph Sketching

Graph sketching in polar coordinates involves plotting points that depend on a radius \( r \) and an angle \( \theta \). Sketching a graph of a polar equation requires understanding how \( r \) changes with \( \theta \). The goal is to plot points for various values of \( \theta \) and observe the pattern they create.

• Begin by selecting key angles (often including \( 0, \pi/2, \pi, 3\pi/2, 2\pi \)).
• For each key angle, calculate the corresponding radius \( r \).

The polar plot emerges as you connect these points with a smooth, continuous line.
For example, in the polar system, the graph of \( r = \cos \theta - 1 \) resembles a cardioid or a heart-shaped curve.
Graph sketching in polar coordinates can reveal beautiful, symmetrical patterns not seen in Cartesian graphs.

Exploring Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in mathematics, especially when dealing with waves, circles, and oscillations. These functions help express relationships involving angles and distances in polar coordinates.

• Cosine is particularly useful in polar equations like \( r = \cos \theta - 1 \) as it determines the value of the radius based on the angle.
• For \( \theta = 0 \) or \( 2\pi \), \( \cos \theta = 1 \), while at \( \theta = \pi \), \( \cos \theta = -1 \).
• Sine and tangent can also provide variations depending on the context of the problem.

In our specific case, \( r = \cos \theta - 1 \) leads to a cardioid shape because the cosine function, being periodic and symmetrical, repeatedly influences the radius to taper and rise over one full cycle of \( \theta \).
Understanding these oscillating values is vital to predicting shapes and patterns in polar graphing.
Recognizing how trigonometric functions affect respect to regular patterns is key to mastering many mathematical applications.