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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 103
Capitolo 1, Problema 103

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle θ\(\theta\) (measured in radians) is A=12r2θA=\(\frac\)12r^2\(\theta\).
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1
Start by understanding the relationship between the area of a circle and the area of a sector. The area of a full circle with radius r is given by the formula A = \(\pi\) r^2.
Recognize that a sector is a portion of the circle, defined by a central angle \( \theta \) (in radians). The fraction of the circle's area that the sector occupies is \( \frac{\theta}{2\pi} \), since the full circle corresponds to an angle of \( 2\pi \) radians.
Calculate the area of the sector by multiplying the fraction of the circle's area by the total area of the circle: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 \).
Simplify the expression: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{\theta r^2}{2} \).
Conclude that the area of the sector is \( A = \frac{1}{2} r^2 \theta \), which is the desired formula.

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Central Angle

A central angle is formed by two radii of a circle that meet at the center. It is measured in degrees or radians, with one full rotation around the circle equating to 360 degrees or 2π radians. Understanding central angles is crucial for calculating the area of a sector, as the area is directly proportional to the angle's measure.
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Trig Values in Quadrants II, III, & IV Example 2

Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and the arc between them. The area of a sector can be calculated using the formula A = (1/2)r²θ, where r is the radius and θ is the central angle in radians. This concept is essential for solving problems related to circular areas and understanding how angles affect the size of the sector.
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Circles in Standard Form

Radians

Radians are a unit of angular measure used in mathematics, particularly in calculus and trigonometry. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Radians provide a natural way to relate angles to arc lengths and areas, making them fundamental for deriving formulas like the area of a sector.
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Converting between Degrees & Radians
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