In this video I go three interesting and useful symmetry properties of polar coordinate functions, which can be used to save time in sketching polar curves. The first property is that if we can replace ϴ by -ϴ and the function remains the same than what we have is a function that is symmetric about the polar axis. The second property is that if the function r is replaced by -r still obtains the same function, than we have is a function symmetric about origin or pole. In other words, we can rotate the function by an angle of π radians and still obtain the same function. The third property is when we replace the angle ϴ with π - ϴ and still update the same function; this means that the curve is symmetric about the angle π/2 radians or 90 degrees.
Also in this video I show how we could have used these properties to save time in graphing the curves from my earlier videos on Examples 6, 7, and 8. This is because those examples involved graphing curves that were indeed symmetric by at least one of the above properties.
This is a very extensive and detailed video illustrating how to use symmetry in polar coordinates to sketch curves, so make sure to watch this video!
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Polar Coordinates: Symmetry
When we sketch polar curves it is sometimes helpful to take advantage of symmetry.
Here are three rules or properties of polar coordinates:
a) If a polar equation is unchanged when θ is replaced by -θ, the curve is symmetric about the polar axis.
b) If the equation is unchanged when r is replaced by -r, or when θ is replaced by θ + π, the curve is symmetric about the pole.
- This means that the curve remains unchanged if we rotate it through 180 degrees about the origin.
c) If the equation is unchanged when θ is replaced by π - θ, the curve is symmetric about the vertical line θ = π/2.
The curves in Examples 6 and 8 are symmetric about the polar axis, since cos(θ) = cos(-θ).
The curves in Example 7 and 8 are symmetric about θ = π/2 because sin(π - θ) = sinθ and cos[2(π - θ)] = cos2θ.
The four-leaved rose, Example 8, is also symmetric about the pole.
These symmetry properties could have been used in sketching the curves.
For instance, in Example 6 we need only have plotted points for 0 ≤ θ ≤ π/2 and then reflected about the polar axis to complete the circle.