Problem 5 - 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. $$s=\frac{\pi}{6}$$

Step-by-Step Solution

Answer

\(s = \frac{\pi}{6}\) represents \(\frac{1}{12}\) of the unit circle's circumference.

Step 1: Understand the Unit Circle

The circumference of the unit circle is given by the formula \(C = 2\pi r\), where \(r = 1\) since it's a unit circle. Therefore, the circumference is \(2\pi\).

Step 2: Express the Given Arc Length

We are given that \(s = \frac{\pi}{6}\). This represents an arc length on the unit circle.

Step 3: Calculate the Fraction of the Circumference

We need to determine what fraction \(\frac{\pi}{6}\) is of the full circumference \(2\pi\). This is done by dividing \(\frac{\pi}{6}\) by \(2\pi\):\[\text{Fraction} = \frac{\frac{\pi}{6}}{2\pi}\]

Step 4: Simplify the Fraction

Simplify the fraction from Step 3:\[\text{Fraction} = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\]Therefore, \(s = \frac{\pi}{6}\) represents \(\frac{1}{12}\) of the circumference of the unit circle.

Key Concepts

Circumference

In the context of a circle, circumference is the total length around its edge. For instance, think of it as the circle's perimeter. A useful way to calculate the circumference of any circle is by using the formula \(C = 2\pi r\), where \(r\) is the radius of the circle.

• The unit circle is a special circle with a radius of 1.
• Thus, the circumference of the unit circle can be calculated as \(2\pi \times 1 = 2\pi\).

This concept is especially important because it allows us to relate any arc length or portion of the circle to the whole, using fractions or percentages.
This understanding forms the basis for problems involving the unit circle that are common in trigonometry and calculus.

Arc Length

An arc length is a portion of the circumference of a circle. You can think of it like a curved line segment on the circle's edge. When we say an arc length is, for example, \(s = \frac{\pi}{6}\), it tells you how much of the circle's edge that particular arc covers.

• It is measured along the circumference, not straight across.
• In the unit circle, the arc length is readily available due to the fixed circumference \(2\pi\).

The emphasis here is on how the arc length relates to the whole, allowing us to express it as a fraction of the circle's total circumference. \Knowing the arc length in terms of \(\pi\) simplifies the realization of the fraction it represents of the total circle.

Fraction Calculation

Calculating the fraction of an arc length compared to the full circumference involves division. If you know the arc length, like \(s = \frac{\pi}{6}\), and you know the full circumference, such as \(2\pi\) for the unit circle, you can find the fraction of the circle's circumference that the arc comprises.
To determine the fraction for \(s = \frac{\pi}{6}\):

• First, set up the division: \(\text{Fraction} = \frac{\frac{\pi}{6}}{2\pi}\).
• Multiply by the reciprocal: \(= \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\).

Therefore, the arc \(\frac{\pi}{6}\) spans \(\frac{1}{12}\) of the unit circle's total circumference. This method reinforces how fractions help comprehend portions of a whole, a skill that proves useful across many mathematical contexts.