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Prove the formula A = ✓ 20 for the area of a sector of a circle with radius r and central angle 0. 2 Hint: Assume 0<

Prove the formula A = ✓ 20 for the area of a sector of a circle with radius r and central angle 0. 2 Hint: Assume 0<

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section: Chapter Questions
Problem 2T
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Prove the formula A = ✓ 20 for the area of a sector of a circle with radius r and central angle 0. 2 Hint: Assume 0<<I and place the center of the circle at the origin so it has the equation x² + y² = r². Then A is the sum of the area of the triangle POQ and the area of the region PQR in the figure below. π 2 Q R x Π Begin by proving the area formula for the case of 0 < 0 < . 2 Consider the construction in the figure above. Since the radius OP has length r in polar coordinates, the area of APQO in polar coordinates is not overlapped by right triangle APQO, is as follows. Since radius OR has length the area of region PQR, that is the area of the area sector -x2 dx area of region PQR = = Ꮎ × Since the area of sector POR is composed of the areas of right triangle APQO and area PQR, we have area of sector POR = area of APQO + area of region PQR as was to be shown.
Transcribed Image Text:Prove the formula A = ✓ 20 for the area of a sector of a circle with radius r and central angle 0. 2 Hint: Assume 0<<I and place the center of the circle at the origin so it has the equation x² + y² = r². Then A is the sum of the area of the triangle POQ and the area of the region PQR in the figure below. π 2 Q R x Π Begin by proving the area formula for the case of 0 < 0 < . 2 Consider the construction in the figure above. Since the radius OP has length r in polar coordinates, the area of APQO in polar coordinates is not overlapped by right triangle APQO, is as follows. Since radius OR has length the area of region PQR, that is the area of the area sector -x2 dx area of region PQR = = Ꮎ × Since the area of sector POR is composed of the areas of right triangle APQO and area PQR, we have area of sector POR = area of APQO + area of region PQR as was to be shown.
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