Problem 135
Question
In calculus we work with real numbers; thus, the measure of an angle must be in radians. The area of a sector of a circle with radius 3 in. and central angle \(\theta\) is \(\frac{3 \pi}{2}\) in. \(^{2} .\) What is the radian measure of \(\theta ?\)
Step-by-Step Solution
Verified Answer
The radian measure of \( \theta \) is \( \frac{\pi}{3} \).
1Step 1: Understand the formula for the area of a sector
The formula to find the area of a sector of a circle is given by \( A = \frac{1}{2} r^2 \theta \), where \( A \) is the area of the sector, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.
2Step 2: Substitute the known values into the formula
Given that the radius \( r = 3 \) inches and the area \( A = \frac{3\pi}{2} \) square inches, substitute these values into the formula: \[ \frac{3\pi}{2} = \frac{1}{2} \times 3^2 \times \theta \].
3Step 3: Simplify the equation
Calculate \( 3^2 \) to get 9, and substitute it into the equation to simplify:\[ \frac{3\pi}{2} = \frac{1}{2} \times 9 \times \theta \] becomes \[ \frac{3\pi}{2} = \frac{9}{2} \theta \].
4Step 4: Solve for \( \theta \)
To solve for \( \theta \), multiply both sides of the equation by the reciprocal of \( \frac{9}{2} \). Therefore: \[ \theta = \frac{3\pi}{2} \times \frac{2}{9} \].
5Step 5: Simplify the expression for \( \theta \)
Multiply the fractions to find \( \theta \):\[ \theta = \frac{3\pi \times 2}{2 \times 9} = \frac{3\pi}{9} = \frac{\pi}{3} \].
Key Concepts
Angle MeasurementSector Area FormulaCentral Angle
Angle Measurement
When it comes to measuring angles, two main units are degrees and radians. In calculus and higher mathematics, radians are often preferred. This is because they relate directly to the geometry of a circle. One full circle around its circumference is equal to an angle of \( 2\pi \) radians. This setup gives us a natural way to measure angles using the radius of a circle.
Understanding radians is simpler if you think about the unit circle. In a unit circle, the radius is always 1. This means the circumference is \( 2\pi \), making \( 2\pi \) radians a complete trip around the circle. Therefore, a quarter of the circle corresponds to \( \frac{\pi}{2} \) radians, and half the circle corresponds to \( \pi \) radians.
The key advantage of radians is how they relate to arc length and sector area in a circle. Unlike degrees, radians make these calculations straightforward due to their intrinsic connection to circle geometry.
Understanding radians is simpler if you think about the unit circle. In a unit circle, the radius is always 1. This means the circumference is \( 2\pi \), making \( 2\pi \) radians a complete trip around the circle. Therefore, a quarter of the circle corresponds to \( \frac{\pi}{2} \) radians, and half the circle corresponds to \( \pi \) radians.
The key advantage of radians is how they relate to arc length and sector area in a circle. Unlike degrees, radians make these calculations straightforward due to their intrinsic connection to circle geometry.
Sector Area Formula
Finding the area of a sector in a circle requires understanding the relationship between the circle's radius and its central angle. A sector can be thought of as a "pizza slice" of the circle, and its area can be calculated using a simple formula:
\[ A = \frac{1}{2} r^2 \theta \]
where \( A \) is the area of the sector, \( r \) is the radius, and \( \theta \) is the central angle in radians. This formula directly stems from the proportion of the circle's overall area that the sector occupies. For a full circle, the area is \( \pi r^2 \). Since a sector is part of this circle, its area depends on the ratio of the central angle \( \theta \) to \( 2\pi \), the full circle.
When you plug in the radius and the central angle in radians to this formula, you can easily calculate the area of the sector. This method is highly effective, especially in calculus, where understanding the geometry helps in solving more complex problems.
\[ A = \frac{1}{2} r^2 \theta \]
where \( A \) is the area of the sector, \( r \) is the radius, and \( \theta \) is the central angle in radians. This formula directly stems from the proportion of the circle's overall area that the sector occupies. For a full circle, the area is \( \pi r^2 \). Since a sector is part of this circle, its area depends on the ratio of the central angle \( \theta \) to \( 2\pi \), the full circle.
When you plug in the radius and the central angle in radians to this formula, you can easily calculate the area of the sector. This method is highly effective, especially in calculus, where understanding the geometry helps in solving more complex problems.
Central Angle
The central angle is the angle subtended at the center of a circle by two radii. It plays a crucial role in determining the geometric properties of a circle's sections, like arc length and sector area. In the context of a circle, the central angle is usually expressed in radians to simplify calculations and interpretations.
The central angle in degrees or radians directly affects the portion of the circle it represents. For example, if the central angle is \( \frac{\pi}{3} \) radians, this angle refers to one-sixth of the full circle, as \( \frac{\pi}{3} \) is a sixth of \( 2\pi \) radians, which is the total angle in a circle.
Understanding the central angle is essential, not just for solving problems involving arcs or sectors, but also in interpreting the circular motion and sinusoidal patterns frequently encountered in physics and engineering. Its measurement determines the size and spread of the circle parts being considered.
The central angle in degrees or radians directly affects the portion of the circle it represents. For example, if the central angle is \( \frac{\pi}{3} \) radians, this angle refers to one-sixth of the full circle, as \( \frac{\pi}{3} \) is a sixth of \( 2\pi \) radians, which is the total angle in a circle.
Understanding the central angle is essential, not just for solving problems involving arcs or sectors, but also in interpreting the circular motion and sinusoidal patterns frequently encountered in physics and engineering. Its measurement determines the size and spread of the circle parts being considered.
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