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Convert the angle from radian measure to degree measure: 5T1/6 Compute the area of the sector with given central angle and radius: e = 270 degrees, r= 4

Convert the angle from radian measure to degree measure: 5T1/6 Compute the area of the sector with given central angle and radius: e = 270 degrees, r= 4

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Angle Conversion and Sector Area Calculation #### Converting Angle from Radian to Degree Measure **Example:** Convert the angle from radian measure to degree measure: \[ \frac{5\pi}{6} \] To convert \(\frac{5\pi}{6}\) radians to degrees, we use the conversion factor \(180^\circ / \pi\): \[ \frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 5 \times 30^\circ = 150^\circ \] So, \(\frac{5\pi}{6}\) radians is equal to \(150^\circ\). --- #### Computing the Area of a Sector **Example:** Compute the area of the sector with given central angle and radius: \[ \Theta = 270^\circ, \quad r = 4 \] The area \(A\) of a sector of a circle is given by the formula: \[ A = \frac{\Theta}{360^\circ} \times \pi r^2 \] Substitute the given values into the formula: \[ A = \frac{270^\circ}{360^\circ} \times \pi \times 4^2 \] Simplify the fraction: \[ A = \frac{3}{4} \times \pi \times 16 \] \[ A = 12\pi \] Therefore, the area of the sector is: \[ 12\pi \, \text{square units} \]
Transcribed Image Text:### Angle Conversion and Sector Area Calculation #### Converting Angle from Radian to Degree Measure **Example:** Convert the angle from radian measure to degree measure: \[ \frac{5\pi}{6} \] To convert \(\frac{5\pi}{6}\) radians to degrees, we use the conversion factor \(180^\circ / \pi\): \[ \frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 5 \times 30^\circ = 150^\circ \] So, \(\frac{5\pi}{6}\) radians is equal to \(150^\circ\). --- #### Computing the Area of a Sector **Example:** Compute the area of the sector with given central angle and radius: \[ \Theta = 270^\circ, \quad r = 4 \] The area \(A\) of a sector of a circle is given by the formula: \[ A = \frac{\Theta}{360^\circ} \times \pi r^2 \] Substitute the given values into the formula: \[ A = \frac{270^\circ}{360^\circ} \times \pi \times 4^2 \] Simplify the fraction: \[ A = \frac{3}{4} \times \pi \times 16 \] \[ A = 12\pi \] Therefore, the area of the sector is: \[ 12\pi \, \text{square units} \]
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