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Chapter 6: Problem 35

Test for symmetry and then graph each polar equation. $$r=\cos \frac{\theta}{2}$$

Short Answer

Expert verified
After testing, the polar equation \(r = cos(胃/2)\) is symmetrical about the vertical line (the polar axis), but not about the pole. The graph of this equation, taking into account its identified symmetry, gives a circle.

Step by step solution

01

Test Symmetry for Vertical Line

Substitute \(胃\) with \(-胃\) in the equation, then simplify. If it produces the same original equation, the graph is symmetric about the vertical line (polar axis). For \(r = cos(胃/2)\), we try to substitute \(胃\) with \(-胃\), we get \(r = cos(-胃/2)\), which is equivalent to the original equation. Therefore, the graph is symmetric about the vertical line.
02

Test Symmetry for Pole

Substitute \(r\) with \(-r\) in the equation, and simplify. If it produces the same original equation, the graph will be symmetric about the pole. In this given equation, we are not able to substitute \(r\) with \(-r\) as \(r\) only appears once in the equation. Therefore, it doesn't satisfy the condition to be symmetric about the pole.
03

Graph the Polar Equation

After determining symmetries, plot points using a range of \(胃\) values from 0 to \(2蟺\), compute corresponding \(r\) values, and then plot these points in polar coordinates. Because \(r = cos(胃/2)\) yields a circular graph, we plot taking into account the identified vertical line symmetry.

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