Problem 9 - 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. $$s=\frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

\(\frac{3\pi}{2}\) represents \(\frac{3}{4}\) of the circumference of the unit circle.

Step 1: Understand the Unit Circle

The unit circle has a radius of 1 and a circumference of \(2\pi\). This circumference is important as it represents a complete revolution around the circle.

Step 2: Recognize the value of s

Given \(s = \frac{3\pi}{2}\). This is the arc length on the unit circle.

Step 3: Find the Fraction of the Circumference

The fraction of the circumference that \(s\) represents is determined by dividing \(s\) by \(2\pi\), the total circumference of the unit circle. Therefore, calculate:\[\frac{\frac{3\pi}{2}}{2\pi}\]

Step 4: Simplify the Fraction

Simplify the expression:\[\frac{\frac{3\pi}{2}}{2\pi} = \frac{3\pi}{2} \times \frac{1}{2\pi} = \frac{3}{4}\]Thus, \(s = \frac{3\pi}{2}\) represents \(\frac{3}{4}\) of the circumference of the unit circle.

Key Concepts

Circumference of a Circle

The circumference of a circle is an essential concept when dealing with circles, especially in the context of the unit circle. The formula to calculate the circumference is given by

• \( C = 2\pi r \)

where \(C\) is the circumference, \(\pi\) is a mathematical constant approximately equal to 3.14159, and \(r\) is the radius of the circle.
In the case of the unit circle, which has a radius of 1, the circumference becomes simply \(2\pi\). This means a full loop around the unit circle measures \(2\pi\) in length.
When trying to understand how much of the circle a given arc length covers, knowing the total circumference is critical. For example, given an arc length \(s\), the fraction it represents of the full circle is \(\frac{s}{2\pi}\).
This fraction tells us precisely how much of the circumference the arc length \(s\) signifies, in relation to the whole circle.

Arc Length

Arc length in the context of circles represents the distance along the circular path. It is like measuring a piece of string that follows the curve of the circle.
For a unit circle, an arc length refers to a segment of the circle's outer boundary, quantified by measurements often expressed in radians. To find the fraction of the circumference that an arc length represents, you can use the formula:

• \( \text{Fraction of circumference} = \frac{\text{Arc length } (s)}{\text{Circumference}} \)

For our specific task, with the arc length \(s = \frac{3\pi}{2}\), we determine its fractional part of the unit circle's total circumference by dividing:

• \( \frac{\frac{3\pi}{2}}{2\pi} = \frac{3}{4} \)

This result indicates that the arc length \(\frac{3\pi}{2}\) covers \(\frac{3}{4}\) of the circle's complete path.
Arc length gives a wonderful perspective on how far along the circle you have traveled relative to the entire circumference.

Radians

Radians are a way to measure angles, particularly suited to circular motion and rotation around circles. Unlike degrees, which segment a circle into 360 parts, radians are derived from the circle's radius.
The entire circumference of the circle corresponds to an angle of \(2\pi\) radians, based on the concept that a circle's radius wrapped around its edge fits \(2\pi\) times.
When you're working with radians, particularly on the unit circle, any angle or arc length expressed in radians corresponds directly to a proportion of the circle's circumference.
For example, \(\pi\) radians cover half the circle (as \(\pi\) out of \(2\pi\) is half), and similarly, \(\frac{3\pi}{2}\) radians, like in our exercise, cover three-quarters of the circle. For any segment of the circle, understanding its radial measure helps us visualize and calculate that fraction with ease.
The radian measure is immensely useful in trigonometry and calculus due to its natural fit with the geometry of circles and its ability to simplify complex calculations involving circular motion.