Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Find the radian and degree measures of the central angle \(\theta\) subtended by the given arc 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

The central angle measures 1.75 radians or approximately 100.21 degrees, and the area of the sector is 14 cm².

Step 1: Calculate the Radian Measure of the Angle

To find the radian measure of the central angle \( \theta \), use the formula: \( \theta = \frac{s}{r} \). Substitute \( s = 7 \, \text{cm} \) and \( r = 4 \, \text{cm} \) into the formula. \( \theta = \frac{7}{4} = 1.75 \, \text{radians} \).

Step 2: Convert Radians to Degrees

To convert from radians to degrees, use the conversion factor \( \frac{180}{\pi} \). Thus, \( \theta_\text{degrees} = 1.75 \times \frac{180}{\pi} \approx 100.21^\circ \).

Step 3: Calculate the Area of the Sector

The area \( A \) of the sector can be found using the formula \( A = \frac{1}{2} r^2 \theta \). Use the radian measure \( \theta = 1.75 \) radians and \( r = 4 \, \text{cm} \). Substitute these into the formula: \( A = \frac{1}{2} \times 4^2 \times 1.75 = 14 \, \text{cm}^2 \).

Key Concepts

Radian Measure

Understanding radian measure is crucial in trigonometry. A radian is a unit of measure for angles. It is based on the radius of a circle. When an arc's length is equal to the circle's radius, the angle in radians at the center of the circle is exactly one radian. This unit is very useful in calculations involving circular motion or periodic phenomena.A key formula to calculate radian measure for a central angle \( \theta \) is:

• \( \theta = \frac{s}{r} \)

where \( s \) is the arc length and \( r \) is the radius.
In our example, with \( s = 7 \text{ cm} \) and \( r = 4 \text{ cm} \), the formula yields:\[\theta = \frac{7}{4} = 1.75 \text{ radians}\]. This simple formula gives a direct relationship between the arc, the angle, and the radius of a circle.
By using this measure, mathematicians and engineers can manipulate angles using algebraic methods.

Degree Conversion

Degrees and radians are both units used to measure angles, but converting between them is a common task in trigonometry. One complete circle is \( 360^\circ \) when measured in degrees and \( 2\pi \) when measured in radians.To convert from radians to degrees, use the conversion factor:

• \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

Using this conversion, an angle \( \theta = 1.75 \text{ radians} \) would be approximately:
\[ \theta_\text{degrees} = 1.75 \times \frac{180}{\pi} \approx 100.21^\circ\]This formula provides an easy way to switch between the metric system (radians) and the imperial system (degrees). It is especially useful when dealing with real-world problems involving rotations and cycles.

Central Angle

The central angle is a fundamental concept in circle geometry. It is the angle subtended by an arc at the center of a circle. The central angle helps determine the size of the arc in relation to the whole circle.For any given circle, the formula to find the central angle in radians is:

• \( \theta = \frac{s}{r} \)

Where \( s \) is the arc length and \( r \) is the radius of the circle.
In our case, substituting \( s = 7 \text{ cm} \) and \( r = 4 \text{ cm} \), we find the central angle:
\[ \theta = \frac{7}{4} = 1.75 \text{ radians} \]The central angle is crucial in determining other properties of the arc, and it provides a direct link between linear dimensions (arc length and radius) and angular dimensions (angle in radians). This concept is widely used in fields such as astronomy, engineering, and navigation.

Sector Area

The sector area is the portion of the total area of a circle that corresponds to a central angle. This part of a circle resembles a "pizza slice" or a "wedge."To calculate the area \( A \) of a sector, use the formula:

• \( A = \frac{1}{2} r^2 \theta \)

where:

• \( r \) is the radius of the circle
• \( \theta \) is the central angle in radians

Using our example values, \( r = 4 \text{ cm} \) and \( \theta = 1.75 \text{ radians} \), the sector area is:
\[ A = \frac{1}{2} \times 4^2 \times 1.75 = 14 \text{ cm}^2 \]This equation derives from the proportion of the angle \( \theta \) out of the full circle \( 2\pi \) radians, making it a simple and effective way to find areas within circular shapes. Understanding the sector area is essential for problems involving segments and angles in various scientific and engineering fields.