Problem 76 - Algebra and Trigonometry

Question

Find the smallest interval for \(\theta\) starting with \(\theta \min =0\) so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it. $$r=4 \sin \theta$$

Step-by-Step Solution

Answer

The smallest interval for \(\theta\) starting at 0 that will graph the function \(r = 4\sin\theta\) without retracing any part of it is [0, 2π].

Step 1: Identify the nature of the function

The function given is \(r = 4\sin\theta\). This is a form of sinusoidal function. In polar coordinates, this usually results in a circle with a radius that is half of the amplitude.

Step 2: Identify the period of the function

The period of \(\sin\theta\) function is 2π. This means the sinusoidal function completes a full cycle over an interval of [0, 2π]. Since our function is \(4\sin\theta\), it will also have a period of 2π; the 4 just increases the amplitude, not the period.

Step 3: Conclude the smallest interval

Given that the sinusoidal function has a period of 2π, the smallest interval for \(\theta\) starting at 0 that will graph the entire function without retracing is the interval [0, 2π]. This interval will show the full circle without retracing any portion of it.

Key Concepts

Understanding Polar Coordinates

Polar coordinates offer a different way of looking at the plane; unlike the rectangular coordinate system which uses x and y coordinates, the polar system defines positions based on a radius and angle. In polar coordinates, each point on the plane is determined by a pair \( (r, \theta) \) where \( r \) is the distance from the origin (commonly known as the pole) and \( \theta \) is the angle measured in radians from the positive x-axis (also termed as the polar axis).

Grasping this concept is essential when graphing equations like \( r = 4\sin\theta \), which may not visually conform to familiar lines or parabolas seen in the Cartesian coordinate system. A point's location is more about how far out and at what angle it is, rather than how far up or down a grid. This radial and angular system is particularly handy when dealing with circular or spiral shapes, where distances from a central point are more relevant than horizontal and vertical positions.

Sinusoidal Functions in Polar Coordinates

Sinusoidal functions are at the heart of periodic phenomena, oscillating between a set of minimum and maximum values. These functions, including \( \sin \theta \) and \( \cos \theta \) in their purest forms, have an amplitude that depicts the height of their peaks and troughs, and a period which is the length of one complete cycle.

In a polar context, the equation \( r = 4\sin\theta \) represents a sinusoidal function. Here, the coefficient 4 amplifies the standard \( \sin \theta \) function, stretching its amplitude, which refers to the maximum radius of the graph in the polar plane. As the angle \( \theta \) increases, this function creates values of \( r \) that oscillate between 0 and 4, depicting a circle with its center at the origin and having a diameter of 4 units. The beauty of sinusoidal functions lies in their predictable nature allowing for the plotting of elegant curves and circles in polar graphs.

Determining the Polar Graph Period

The period of a graph in polar coordinates, as in other contexts, is the smallest interval after which the function's values start to repeat. For a pure sine or cosine function, this period is \( 2\pi \), which corresponds to one complete rotation around the circle or 360 degrees.

Our example \( r = 4\sin\theta \) inherits the same period as a standard sine function because the multiplier affects amplitude, not period. Thus, when graphing this function, you only need to sketch it from \( \theta = 0 \) to \( \theta = 2\pi \) to capture its entire essence without redundancy. During this interval, \( \theta \) sweeps out a full circle, and since all possible values of \( r \) for this function occur within this range, we obtain a complete graph. This understanding ensures that the curve is drawn once entirely without retracing, leading to a clear and accurate representation of the function in polar form.