Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Determine what fraction of the circumference of the unit circle each value of s represents. For example, \(s=\pi\) represents \(\frac{1}{2}\) of the circumference of the unit circle. Do not use a calculator. $$s=\frac{5 \pi}{2}$$

Step-by-Step Solution

Answer

\(s = \frac{5\pi}{2}\) represents \(\frac{5}{4}\) of the unit circle's circumference, meaning it goes around once fully and then another quarter.

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1. The circumference of a circle is calculated using the formula \(2\pi r\). For the unit circle, where \(r = 1\), the circumference is \(2\pi\). Our task is to find out what fraction of this total circumference \(2\pi\) is represented by \(s = \frac{5\pi}{2}\).

Step 2: Determine the Fraction of Circumference

To find what fraction \(s\) represents, divide the given arc length \(s\) by the total circumference. This is given by the formula: \[ \text{Fraction} = \frac{s}{2\pi} \] For \(s = \frac{5\pi}{2}\), substitute to get: \[ \frac{\frac{5\pi}{2}}{2\pi} \]

Step 3: Simplify the Fraction

Simplify the expression \(\frac{\frac{5\pi}{2}}{2\pi}\):1. First, rewrite it as \(\frac{5\pi}{2} \times \frac{1}{2\pi}\).2. Cancel out the \(\pi\) from the numerator and the denominator.3. This leaves us with \(\frac{5}{2} \times \frac{1}{2} = \frac{5}{4}\).This means \(\frac{5\pi}{2}\) represents \(\frac{5}{4}\) of the unit circle's circumference.

Key Concepts

Circumference

The circumference of a circle is a fundamental concept in geometry, representing the total distance around the circle. For any circle, it is calculated using the formula:

• \(C = 2\pi r\)

Here, \(r\) is the radius of the circle, and \(\pi\) (pi) is a constant approximately equal to 3.14159. For the unit circle, which specifically has a radius of 1, the formula simplifies considerably. Plugging \(r = 1\) into the formula results in a circumference of \(2\pi\).
This means the total distance around the unit circle is \(2\pi\). Understanding this helps in determining what fraction any given arc length represents of the whole circle.
The circumference being \(2\pi\) serves as the baseline for our calculations when dealing with arcs and sectors on the unit circle.

Arc Length

The concept of arc length is intimately connected with both the size of a circle and the central angle that subtends the arc. An arc is essentially a section of the circumference of a circle. In the context of a unit circle, its arc length can be precisely evaluated using the same formulas you'd use for a larger circle.
The arc length formula for any circle is:

• \(s = r \theta\)

Where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) (theta) is the central angle in radians. For the unit circle (where \(r = 1\)), this simplifies to \(s = \theta\).
This means, for example, if \(\theta = \frac{\pi}{2}\), then the arc length \(s\) is also \(\frac{\pi}{2}\). For situations like the one in the exercise, where \(s = \frac{5\pi}{2}\), it is useful to understand that \(\theta\) is already represented by \(\frac{5\pi}{2}\).
Knowing the arc length allows us to calculate further details about the segment of the circumference it covers.

Fraction of Circle

Determining what fraction an arc represents of the entire circumference is a vital step in understanding its comparative length. This process involves a straightforward calculation: dividing the arc length by the total circumference. Using the unit circle's properties makes this method especially clear.
Given our arc length \(s = \frac{5\pi}{2}\) from the exercise, the fraction of the circle it covers is given by:

• \(\text{Fraction} = \frac{s}{2\pi}\)

Substitute in \(\frac{5\pi}{2}\) for \(s\), and we get:

• \(\frac{\frac{5\pi}{2}}{2\pi} = \frac{5\pi}{2} \times \frac{1}{2\pi} = \frac{5}{4}\)

This calculation shows that an arc length of \(\frac{5\pi}{2}\) is greater than the full circumference, representing \(\frac{5}{4}\) or \(1\frac{1}{4}\) portions of the full circle. Understanding the fraction of a circle not only helps in precise mathematical analysis but also enriches visual intuition about the size of angles and arcs in circular motion.