While both sine and cosine functions create beautiful and intricate polar graphs, there are a few key differences that can help you distinguish between them.
Visual Clues:
- Symmetry:
- Sine graphs: Often exhibit symmetry about the y-axis. This means that if you fold the graph along the vertical axis, the two halves would match.
- Cosine graphs: Often exhibit symmetry about the x-axis. Folding the graph along the horizontal axis would result in identical halves.
- Starting Point:
- Sine graphs: Typically start at the origin (0, 0) and then move upward or downward.
- Cosine graphs: Usually start at a maximum or minimum point on the x-axis and then oscillate.
Equations:
- General Form: Both sine and cosine graphs can be represented by the general equation: r = a ± b sin(θ) or r = a ± b cos(θ), where a and b are constants.
- Key Difference: The presence of sin(θ) usually indicates a sine graph, while cos(θ) indicates a cosine graph.
Examples:
- r = 2 sin(θ): This graph is a circle with a radius of 1, centered at (1, 0) on the y-axis. It exhibits symmetry about the y-axis and starts at the origin.
- r = 3 cos(θ): This graph is a circle with a radius of 1.5, centered at (1.5, 0) on the x-axis. It shows symmetry about the x-axis and starts at a maximum point on the x-axis.
Remember that these are general guidelines. The specific shape of the graph can be influenced by the values of a and b in the equation. However, by looking for these key characteristics, you can often determine whether a polar graph is primarily based on sine or cosine functions.
