True–False Determine whether the statement is true or false....

True–False Determine whether the statement is true or false. Explain your answer.
$$
\begin{array}{l}{\text { The } x \text { -axis is tangent to the polar curve } r=\cos (\theta / 2) \text { at }} \\ {\theta=3 \pi .}\end{array}
$$

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Key Concepts

Polar Coordinates

Polar coordinates provide a method of describing points in the plane using a radius and an angle. In this system, curves can be expressed as equations of the form r = f(?), which can offer insights into the symmetry and periodicity of the curve. This concept is foundational when analyzing curves that are naturally defined with respect to a central point rather than using the Cartesian x and y axes.

Tangent Lines to Curves

A tangent line to a curve at a given point is a straight line that touches the curve without crossing it, representing the instantaneous direction of the curve at that point. In polar coordinates, determining whether a specific line (like the x-axis) is tangent to a curve involves analyzing how the curve behaves in the vicinity of a specific angle, using differentiation techniques or by converting the polar equation into the Cartesian form.

Conversion between Polar and Cartesian Coordinates

Converting from polar to Cartesian coordinates is crucial for examining the geometric properties of polar curves in a familiar algebraic framework. The formulas x = r cos ? and y = r sin ? are used to translate polar equations into Cartesian equations, which can then be used to study aspects like the slope of the tangent line and to confirm tangency conditions with respect to a given line such as the x-axis.

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