sec.%2010.8 - CHAT Pre-Calculus Section 10.8 1 Graphs of...

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Unformatted text preview: CHAT Pre-Calculus Section 10.8 1 Graphs of Polar Equations To begin graphing in the polar coordinate system we will start with plotting points. Look at the polar equation  sin 4  r . Make a table. Graph the points and then connect them. θ r 6  2 3  5 . 3 3 2  2  4 3 2  5 . 3 3 2  6 5  2  6 7  2  2 3  4  6 11  2   2 2   1 2 3 2 3  4 Notice that the points from π to 2 π retrace what is already graphed. CHAT Pre-Calculus Section 10.8 2 Symmetry Just as symmetry helps us to graph equations in rectangular form, it also helps us to graph in polar form. The graph above shows symmetry with respect to the y- axis. But in polar coordinates the y-axis is the line 2    . In general, we have 3 types of symmetry for polar graphs. 2   2 3  2   2 3  2   2 3  Symmetry with respect to the line 2    Symmetry with respect to the polar axis Symmetry with respect to the pole. CHAT Pre-Calculus Section 10.8 3 You will have to refer back to the sum and difference formulas from chapter 5: v u v u v u v u v u v u v u v u v u v u v u v u sin sin cos cos ) cos( sin sin cos cos ) cos( sin cos cos sin ) sin( sin cos cos sin ) sin(             v u v u v u v u v u v u tan tan 1 tan tan ) tan( tan tan 1 tan tan ) tan(         Tests for Symmetry in Polar Coordinates The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation. 1. The line 2    : Replace ) , (  r with ) , (    r or ) , (    r 2. The polar axis : Replace ) , (  r with ) , (   r or ) , (     r 3. The pole : Replace ) , (  r with ) , (    r or ) , (  r  CHAT Pre-Calculus Section 10.8 4 You may also need to refer back to the Odd/Even identities of chapter 4: cos(-θ) = cos(θ) sec(-θ) = sec(θ) sin(-θ) = -sin(θ) csc(-θ) = -csc(θ) tan(-θ) = -tan(θ) cot(-θ) = -cot(θ) Example : Describe the symmetry of the polar equation ) sin 1 ( 2    r . Solution : Test for each type of symmetry. 1. The line 2    : Replace ) , (  r with ) , (    r or ) , (    r . Let’s pick ) , (    r ....
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sec.%2010.8 - CHAT Pre-Calculus Section 10.8 1 Graphs of...

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