Problem 106 - Precalculus : Building Concepts and Connections

Question

What is the area of the portion of the unit circle swept Sut by an angle of \(\frac{\pi}{6}\) radians?

Step-by-Step Solution

Answer

The area of the portion of the unit circle swept out by an angle of \(\pi/6\) radians is \(\pi/12\) square units.

Step 1: Understanding the Problem and Concepts Involved

The problem requires finding the area of a portion of a unit circle, referred to as a sector. A sector is a pie slice shaped portion of a circle. The area for a sector is calculated using the formula \[Area = \frac{1}{2} r^2\theta\], where 'r' is the radius of the circle and '\(\theta\)' is the angle in radians. Given that the radius here is 1, the formula simplifies to \[Area = \frac{1}{2}\theta\].

Step 2: Substitute the Given Value of the Angle

Next, substitute the given value of the angle (\(\pi/6\) rad) into the simplified formula. This results in \[Area = \frac{1}{2} \cdot \frac{\pi}{6}\].

Step 3: Calculate the Area

After substituting the value, compute the area. When calculating, the result is \[Area=\frac{\pi}{12}\].

Key Concepts

Radian Measure

Understanding radian measure is crucial when working with circles and angles in trigonometry. It's a way of expressing angles based on the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. Unlike degrees, which are based on dividing a circle into 360 arbitrary units, radians provide a direct relationship between the angle and the arc it subtends, which makes calculations more natural and simpler in mathematics.

• The circumference of a unit circle (radius = 1) is \(2\pi\), meaning there are \(2\pi\) radians in a full turn.
• A straight angle is \(\pi\) radians, whereas a right angle is \(\pi/2\) radians.
• To convert from degrees to radians, we multiply by \(\pi/180\).
• Conversely, we convert from radians to degrees by multiplying by \(180/\pi\).

Understanding radians allows you to work with angles in a way that relates directly to the geometry of a circle, which is particularly useful in trigonometry and when dealing with periodic functions like sine and cosine.

Sector Area Calculation

The process of finding the area of a sector is an essential application of radian measure in geometry. A sector is a 'pie-shaped' portion of a circle, and its area can be thought of as a fraction of the entire circle's area proportional to the angle the sector spans. The formula for this calculation reflects the swept angle in radians:
\[Area = \frac{1}{2} r^2\theta\]
Where:\

• \(r\) represents the radius of the circle,
• \(\theta\) denotes the central angle in radians.

For a unit circle where \(r = 1\), the formula simplifies to \(Area = \frac{1}{2}\theta\) as the radius squared becomes 1. This simplicity is why unit circles are so valuable in problems involving sectors and radian measures. By substituting the measure of the angle in radians into the formula, you determine the area of the sector as a portion of the whole circle's area.

Trigonometry in Precalculus

In precalculus, trigonometry provides a foundation for understanding the relationships between the sides and angles of triangles, and by extension, the relationships within circles. It involves studying functions such as sine, cosine, and tangent, which are essential when analyzing periodic phenomena.

• Trigonometric identities illustrate the properties and relationships between these functions and are vital in simplifying expressions and solving equations.
• The unit circle is a valuable tool for visualizing trigonometric functions and their values at different angles, especially when those angles are given in radian measure.
• A deep comprehension of how to calculate arc lengths, sector areas, and solve trigonometric equations forms an integral part of trigonometry in precalculus.

By integrating the radian measure concept and sector area calculations, students gain a more thorough understanding of how these topics interconnect, setting a strong groundwork for calculus and further studies in mathematics.