Problem 7 - 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. $$s=3 \pi$$

Step-by-Step Solution

Answer

The value of \(s = 3\pi\) represents \(\frac{3}{2}\) of the circumference of the unit circle.

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 \(C = 2\pi r\). For a unit circle where \(r = 1\), the circumference \(C = 2\pi\).

Step 2: Identify the Given Arc Length

The problem provides \(s = 3\pi\), which represents the arc length on the unit circle. This arc length is part of the total circumference \(2\pi\).

Step 3: Calculate the Fraction of the Circumference

To determine what fraction of the circumference \(s\) represents, divide the arc length \(s = 3\pi\) by the total circumference \(2\pi\). This gives:\[\text{Fraction} = \frac{3\pi}{2\pi}\]

Step 4: Simplify the Fraction

Simplify the fraction by canceling out \(\pi\) in the numerator and denominator:\[\frac{3\pi}{2\pi} = \frac{3}{2}\]

Step 5: Interpret the Result

The simplified fraction \(\frac{3}{2}\) indicates that \(s = 3\pi\) represents \(\frac{3}{2}\) or 1.5 times the circumference of the unit circle.

Key Concepts

Circumference

Circumference is a fundamental concept when dealing with circles. It refers to the distance around a circle, similar to the perimeter of a polygon. You calculate the circumference using the formula \( C = 2\pi r \), where \( r \) is the radius of the circle.
On the unit circle, which has a radius of 1, this formula simplifies to \( C = 2\pi \). This means the total distance around the unit circle is \( 2\pi \). The unit circle's uniformity makes it an ideal model for understanding trigonometry and geometry concepts. Understanding circumference helps you find arc lengths and solve related math problems.

Arc Length

An arc is a portion of a circle's circumference. Arc length is the measure of this portion and can be found by remembering that the full circle's arc is the circumference. In our specific example of the unit circle, if the given arc length \( s = 3\pi \), it extends beyond one complete lap around the circle.
The length of an arc is found by a fraction of the full circle's circumference depending on the angle or the assigned arc itself.
To find out how much of the circle the arc covers, compare the arc length to the full circumference. This comparison tells us what fraction of the circle it represents. Calculating and understanding arc length is crucial in trigonometry, as it links angles and the geometry of circles.

Fraction

Fractions represent parts of a whole, and in this case, describe parts of the circle's circumference. The problem requires expressing the arc length, \( s = 3\pi \), as a fraction over the circle's total circumference \( 2\pi \).
To do this, divide the arc length by the total circumference:

• \( \text{Fraction} = \frac{3\pi}{2\pi} \)

This equation allows cancellation of \( \pi \) from the numerator and denominator, simplifying to \( \frac{3}{2} \). This means that the arc accounts for \(1.5\) times the full circumference of the unit circle. Understanding fractions in this context shows the relationship between parts of a circle and its entirety.

Radius

The radius of a circle is the distance from its center to any point along its circumference. In the unit circle, the radius is always 1. This standard measurement simplifies many equations, such as calculating the circumference \( C = 2\pi \cdot 1 \), meaning the circle has a circumference of \( 2\pi \).
The radius is also essential in defining properties of a circle in coordinate geometry and in the broader context of circular motion and trigonometry.
Knowing the radius helps establish the size and scale of other features within the circle. For instance, when the arc length or parts of the circle are expressed or measured, understanding the radius directly influences how these dimensions are perceived and calculated.