Problem 11 - 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. $$s=\frac{5 \pi}{4}$$

Step-by-Step Solution

Answer

\(s = \frac{5\pi}{4}\) is \(\frac{5}{8}\) of the circumference.

Step 1: Understand the Relationship between s and Circumference

To determine the fraction of the circle represented by a given arc length, we need to know that the circumference of the unit circle is given by the formula \(C = 2\pi\). The circle is divided into equal arcs based on multiples of \(\pi\).

Step 2: Express s in Terms of Circumference

The fraction of the circumference a given \(s\) represents can be expressed as the ratio \(\frac{s}{2\pi}\). Thus, for \(s = \frac{5\pi}{4}\), this becomes \(\frac{\frac{5\pi}{4}}{2\pi}\).

Step 3: Simplify the Fraction

Simplify \(\frac{\frac{5\pi}{4}}{2\pi}\) by canceling \(\pi\) from both the numerator and the denominator, resulting in \(\frac{5}{8}\). Hence, \(s = \frac{5\pi}{4}\) represents \(\frac{5}{8}\) of the circumference of the unit circle.

Key Concepts

Circumference

The circumference of a circle is the distance around it. For a circle with radius 1, known as the unit circle, this distance is particularly simple to calculate. The formula for the circumference of any circle is given by \( C = 2\pi r \), where \( r \) is the radius. Since the radius of the unit circle is 1, the formula becomes \( C = 2\pi \).

Hence, the circumference of the unit circle is \(2\pi\). This means if you traveled all the way around the unit circle, you'd cover a distance of \(2\pi\) units. Understanding this is crucial when working with arc lengths and determining the fraction of the circle they represent.

Arc Length

An arc length is a portion of the circumference of a circle. To find what fraction of the entire circle an arc length represents, you put the length of the arc over the total circumference. For example, given an arc length \( s \) on the unit circle, the fraction of the circumference it covers is \( \frac{s}{2\pi} \).

This formula helps us understand that the arc length is directly proportional to the fraction of the circle it represents. For instance, if \( s = \frac{5\pi}{4} \), the process to find the fraction of the unit circle's circumference it represents involves simplifying the expression \( \frac{\frac{5\pi}{4}}{2\pi} \). It turns out that this arc length represents \( \frac{5}{8} \) of the circle. This approach works for any arc length \( s \) and is grounded in knowing the full circumference is always \(2\pi\).

Fractions

Fractions represent parts of a whole. When working with the unit circle, we're often interested in what fraction of the circle a certain arc length represents. To convey this idea mathematically, we use fractions to compare the arc length \( s \) to the entire circumference \(2\pi\).

In the exercise, we simplified the fraction \( \frac{\frac{5\pi}{4}}{2\pi} \) to determine what part of the circle \( \frac{5\pi}{4} \) represents. By simplifying and canceling out \( \pi \) from both the numerator and denominator, we arrive at \( \frac{5}{8} \). This simplified fraction shows that the arc length is five-eighths of the total circumference.

Understanding fractions in this context is vital for making comparisons and analyzing portions of shapes, especially circles in trigonometry.

Trigonometry

Trigonometry is a branch of mathematics dealing with relationships between angles and sides of triangles, but it also has a deep connection with circles, especially the unit circle. The unit circle is a key concept in trigonometry because it helps us understand sine, cosine, and other trigonometric functions.

Knowing the circumference of the unit circle allows us to relate arc lengths to angles. For an arc measuring \( \theta \) radians, its length can be found using the formula \( s = r\theta \). In the unit circle, where \( r = 1 \), this simplifies to \( s = \theta \). This relationship shows how angles in radians are just another way to measure arc lengths on the unit circle.

By mastering the unit circle and its circumference, students can better grasp how trigonometric functions relate to geometric concepts, providing a strong foundation for advanced math topics.