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Polar coordinates are a way to describe the position of a point in a plane using the distance from a reference point and the angle from a reference direction. In polar coordinates, we often use the symbols \( (r, \theta) \), where \( r \) is the radial distance from the origin (or pole) and \( \theta \) is the angular parameter measured in radians.
In this exercise, we have a polar curve described by \( r=3\sin \theta \). Here, the value of \( r \) depends on the angle \( \theta \), and it varies as \( \theta \) changes. This particular function traces out a circle when fully plotted.
Polar coordinates are especially useful when dealing with problems exhibiting circular or rotational symmetry, allowing for more straightforward calculations and a clear visual representation in terms of angles and radii.

Derivatives are fundamental tools in calculus allowing us to understand the rate at which things change. In the context of polar coordinates, taking the derivative of a polar function like \( r = 3\sin \theta \) involves differentiating with respect to \( \theta \).

• The derivative of \( r \) with respect to \( \theta \), noted as \( r' \), tells us how the radius changes as the angle changes.
• In this exercise, \( r' = 3\cos \theta \) is the derivative.

To find the slope of a tangent line at a particular point, say when \( \theta = \frac{\pi}{2} \), you substitute this value into the derivative to find \( r' \) at that point. Understanding derivatives in polar coordinates helps us find tangents and analyze curves.

Trigonometric functions, including sine and cosine, are crucial in relating angles to sides of right triangles, and they play a central role in polar coordinates. The function \( \sin \theta \) is periodic and oscillates between -1 and 1, dictating the value of \( r \) when used in polar equations like \( r = 3\sin \theta \).
In this problem, \( \theta = \frac{\pi}{2} \) leads to \( \sin \left( \frac{\pi}{2} \right) = 1 \), resulting in \( r = 3 \). Trigonometric identities and properties are valuable as they help in simplifying the function and making sense of the curve's behavior.
The tangent line at any point on a polar curve can also involve trigonometric functions to express its slope effectively. Here, the relation becomes clear in solutions where the horizontal slope or other characteristics depend on these foundational trigonometric values.