Problem 33 - Precalculus: Functions and Graphs

Question

Exer. 33-34: (a) Find the radian and degree measures of the central angle \(\boldsymbol{\theta}\) subtended by the given are of length \(s\) on a circle of radius \(r .\) (b) Find the area of the sector determined by \(\theta\) $$s=7 \mathrm{cm}, \quad r=4 \mathrm{cm}$$

Step-by-Step Solution

Answer

(a) \(\theta = \frac{7}{4}\) radians, \(100.53^\circ\); (b) Area = 14 cm².

Step 1: Understand the Relationship Between Arc Length, Radius, and Radians

To find the radian measure of the central angle \(\theta\), recall the formula for arc length:\(s = r \theta\). Here, \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is given in radians.

Step 2: Calculate the Radian Measure of \(\theta\)

Using the formula \(s = r \theta\), plug in the given values: \(7 = 4 \theta\). Solve for \(\theta\): \(\theta = \frac{7}{4}\) radians.

Step 3: Convert Radian Measure to Degrees

To convert radians to degrees, use the conversion factor \(\frac{180}{\pi}\). So, \(\theta\) in degrees is \(\frac{7}{4} \times \frac{180}{\pi} \approx 100.53^\circ\).

Step 4: Use \(\theta\) to Determine the Area of the Sector

The area \(A\) of the sector is given by the formula \(A = \frac{1}{2} r^2 \theta\), where \(\theta\) is in radians. Substitute the known values: \(A = \frac{1}{2} \times 4^2 \times \frac{7}{4}\).

Step 5: Calculate the Area of the Sector

Perform the arithmetic: \(A = \frac{1}{2} \times 16 \times \frac{7}{4} = \frac{28}{2} = 14\). So, the area of the sector is 14 square centimeters.

Key Concepts

Radian Measure

When dealing with circles, the concept of a radian is fundamental. It is an alternate way to measure angles instead of the more familiar degree measurement. One radian is defined as the angle subtended by an arc that has a length equal to the circle's radius. To find the radian measure of an angle, you use the formula for arc length: \(s = r\theta\). Here, \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the angle in radians. In the given problem, with an arc length of 7 cm and a radius of 4 cm, you rearrange the formula to solve for \(\theta\):

• \(\theta = \frac{s}{r} = \frac{7}{4}\) radians.

This allows us to understand that radians provide a direct relationship between arc lengths and radii.

Degree Conversion

Though radians are essential in mathematics, degrees are often more intuitive since they are part of everyday language. Degrees can be converted from radians using the fixed ratio \(\frac{180}{\pi}\), because one full circle is \(2\pi\) radians or 360 degrees. The conversion formula is:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Using our example where \(\theta = \frac{7}{4}\) radians, the degree measure is:

• \(\frac{7}{4} \times \frac{180}{\pi} \approx 100.53^\circ\).

This conversion allows you to communicate the size of an angle more easily in scenarios where degrees are used.

Arc Length

Arc length is a segment of the circumference of a circle. It's directly related to the radian measure of the central angle and the radius of the circle. The arc length formula \(s = r\theta\) helps link both of these elements. Consider this – when the angle \(\theta\) is measured in radians, the arc length \(s\) provides a sense of how much of the circle's outline is covered by the angle. In our example:

• The arc length was given as 7 cm.
• The radius was 4 cm.

This relationship helps in calculating the radian measure, which ties back to how far the angle "stretches" along the circle's edge.

Sector Area

The concept of sector area comes into play when you want to find out the space enclosed by two radii and the arc between them. The formula for calculating the area of a sector is:\[A = \frac{1}{2} r^2 \theta\]where \(A\) is the area, \(r\) is the radius, and \(\theta\) is the angle in radians. To determine the area of the sector in our problem:

• We use \(\theta = \frac{7}{4}\) radians and \(r = 4\) cm.
• The calculation is \(A = \frac{1}{2} \times 4^2 \times \frac{7}{4} = 14\) square centimeters.

Understanding these calculations helps in visualizing how much of the circle's surface is contained within the two radii and the arc.