Problem 7 - Calculus Early Transcendentals

Question

Explain three symmetries in polar graphs and how they are detected in equations.

Step-by-Step Solution

Answer

In summary, there are three types of symmetry in polar graphs: 1. Symmetry about the polar axis (x-axis) - achieved by replacing θ with -θ and checking if the equation remains unchanged. 2. Symmetry about the line θ = π/2 (y-axis) - achieved by replacing θ with (π - θ) and checking if the equation remains unchanged. 3. Symmetry about the pole (origin) - achieved by simultaneously replacing r with -r and θ with (θ + π), and checking if the equation remains unchanged. Polar equations can exhibit one or more types of symmetry, which help simplify their analysis and graphing, and provide a deeper understanding of their properties.

Step 1: 1. Symmetry about the polar axis (x-axis)

To check for symmetry about the polar axis in a polar equation, we should replace θ with -θ and see if the equation remains unchanged. If the equation remains the same, the graph of the equation exhibits symmetry about the x-axis. For instance, let r(θ) = 3cos(θ) be a polar equation. To check for symmetry, we can replace θ with -θ: r(-θ) = 3cos(-θ) r(-θ) = 3cos(θ) (since cosine is an even function) Since the equation remains unchanged, 3cos(θ) is symmetric about the polar axis.

Step 2: 2. Symmetry about the line θ = π/2 (y-axis)

To check for symmetry about the line θ = π/2 (y-axis) in a polar equation, we need to replace θ with (π - θ) and see if the equation remains the same. If the equation remains the same, the graph of the equation displays symmetry about θ = π/2. For example, let r(θ) = 2sin(θ) be a polar equation. To check for symmetry, we can replace θ with (π - θ): r(π - θ) = 2sin(π - θ) r(π - θ) = 2sin(θ) (since sine is an odd function) Since the equation remains unchanged, 2sin(θ) is symmetric about θ = π/2.

Step 3: 3. Symmetry about the pole (origin)

To check for symmetry about the pole (origin) in a polar equation, we should replace r with -r and θ with (θ + π) simultaneously, and see if the equation stays the same. If the equation remains the same, the graph of the equation has symmetry about the pole. For instance, let r(θ) = 4sin(2θ) be a polar equation. To check for symmetry, we can replace r with -r and θ with (θ + π): r(θ + π) = 4sin(2(θ + π)) r(θ + π) = 4sin(2θ + 2π) Now, applying the property of sine, sin(x + 2π) = sin(x), we find that: r(θ + π) = 4sin(2θ) Since replacing r with -r and θ with (θ + π) results in the same equation as the original, the polar equation 4sin(2θ) is symmetric about the pole.