Test for symmetry with respect to a. the polar axis. b. the line θ=(π)/(2) c. the pole. r=sinθ

Name: Problem 7, Chapter 7: Algebra and Trigonometry
Uploaded: 2020-06-11T07:43:35-08:00
Duration: 1 min 7 s
Channel: Ankit Gupta
Description: Test for symmetry with respect to a. the polar axis. b. the line θ=(π)/(2) c. the pole. r=sinθ

step 1 We are given equation r is equal to sine theta. We will check its symmetry along the polar axis

Test for symmetry with respect to a. the polar axis. b. the...

Key Concepts

Polar Coordinate System

The polar coordinate system expresses points in the plane using a radius and an angle. This system is particularly useful for describing curves that have circular or rotational qualities and involves the use of 'r' (the distance from the origin) and '?' (the angle from the positive x-axis).

Symmetry in Polar Coordinates

Symmetry in polar coordinates refers to situations where a polar curve is invariant under specific transformations, such as reflections about the polar axis, the line ? = ?/2, or the pole. Recognizing these symmetries can simplify the process of graphing the curve and understanding its properties.

Symmetry with Respect to the Polar Axis

Testing for symmetry with respect to the polar axis involves substituting -? for ? in the polar equation. If the resulting equation is equivalent to the original, the curve is symmetric about the polar axis. This concept is important for identifying mirror images across the horizontal line through the origin in polar plots.

Symmetry with Respect to the Line ? = ?/2

For symmetry about the line ? = ?/2, the method typically requires replacing ? with ? - ? or using appropriate transformations that reflect the curve over the vertical line through the origin. If the equation remains unchanged after the substitution, the curve exhibits this type of symmetry. Understanding this helps in recognizing vertical symmetry in the graph.

Symmetry with Respect to the Pole

Testing for symmetry with respect to the pole involves checking whether replacing r with -r produces an equation equivalent to the original. This test is crucial because many polar curves, though they might appear different when r is negative, reflect symmetric properties with respect to the origin (pole) when the negative radius is appropriately interpreted.

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