Problem 4 - 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. $$s=\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

\(s = \frac{3\pi}{4}\) represents \(\frac{3}{8}\) of the unit circle's circumference.

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1. The circumference of a circle is calculated as the radius multiplied by \(2\pi\). Therefore, the circumference of the unit circle is \(2\pi\).

Step 2: Expressing s in Terms of Circumference

The value of \(s\) given is \(\frac{3\pi}{4}\). We need to determine what fraction of the unit circle's circumference this represents. Since the circumference is \(2\pi\), we express this fraction as \(\frac{3\pi}{4} / 2\pi\).

Step 3: Calculating the Fraction

To find the fraction, divide the expression by cancelling \(\pi\) from both the numerator and the denominator: \[\frac{3\pi}{4} \div 2\pi = \frac{3\pi}{4} \times \frac{1}{2\pi} = \frac{3}{8}.\]

Step 4: Comparing with the Unit Circle

The value \(s = \frac{3\pi}{4}\) represents \(\frac{3}{8}\) of the circumference of the unit circle. This essentially tells us how far around the circle we are traveling if we start from the point \((1,0)\) on the x-axis and move counterclockwise.

Key Concepts

Circumference

The circumference of a circle is the total distance around it. For any circle, the formula to find circumference is given by multiplying the radius by \(2\pi\). In the case of the unit circle, which has a radius of 1, this makes the calculation straightforward:

• Formula: Circumference = Radius \( \times 2\pi \)
• For a unit circle: Circumference = \(1 \times 2\pi = 2\pi\)

Thus, the unit circle has a circumference of \(2\pi\), meaning if you went all the way around, you'd have traveled \(2\pi\) units. When dealing with fractions involving \(\pi\), knowing the circumference helps in understanding how much of the circle's boundary is being considered.

Radians

Radians are a unit of measure for angles, based on the radius of the circle. One complete rotation around a circle is \(2\pi\) radians. Here’s why radians are useful:

• They relate directly to the circle's circumferential distance.
• A radian is the angle created when the radius is wrapped along the circle's edge.

So in terms of degrees, \(2\pi\) radians equal 360 degrees, meaning \(\pi\) radians cover half the circle, or 180 degrees. Using radians helps us consider portions of the circle more naturally, especially in the context of the unit circle.To express \(s = \frac{3\pi}{4}\) in radians:

• The angle \(\frac{3\pi}{4}\) is 3/4 of the way to \(\pi\), marking a significant segment of the circle.

This value links directly to understanding what fraction of the circumference is described, as well as how far around the circle you have rotated.

Fraction of Circle

When approaching the problem of finding out what fraction of the unit circle's circumference a given value represents, we start by recognizing the full circumference is \(2\pi\). Any value of \(s\) can thus be understood in terms of its proportion of \(2\pi\).Taking the given value:

• \(s = \frac{3\pi}{4}\)
• Calculate what fraction of the circumference it covers: divide \(\frac{3\pi}{4}\) by \(2\pi\).
• This results in: \[\frac{3\pi}{4} \div 2\pi = \frac{3}{8}\]

This \ means \(s\) represents \(\frac{3}{8}\) of the full circle circumference. This is akin to traveling 3/8ths of the way around the circle, starting from the point (1,0). Understanding fractions of the circle is essential for linking linear arc lengths with angular measure, especially when dealing with rotational movement or trigonometric functions.