390 CHAPTER 6 Trigonometric Functions 5 Find the Area of a Sector of a Circle Consider a circle of radius r. Suppose that ,θ measured in radians, is a central angle of this circle. See Figure 16. We seek a formula for the area A of the sector (shown in blue) formed by the angle .θ Consider a circle of radius r and two central angles θ and ,1θ both measured in radians. See Figure 17. From geometry, the ratio of the measures of the angles equals the ratio of the corresponding areas of the sectors formed by these angles. That is, A A 1 1 θ θ = Now suppose that 2 radians. 1θ π = Then A area 1 = of the r circle . 2 π = Solving for A, we find A A r r 2 1 2 1 1 2 2 θ θ π θ π θ = = = ↑ A r ; 2 1 2 1 π θ π = = Figure 17 θ θ = A A 1 1 A1 u1 A u r Figure 16 Sector of a Circle A u r NOTE Again, as with the arc length formula, the Area of a Sector formula requires that the angle θ be expressed in radians. Then the unit used for the radius is squared, and the area is given in square units. j THEOREM Area of a Sector The area A of the sector of a circle of radius r formed by a central angle of θ radians is A r 1 2 2θ = (8) 6 Find the Linear Speed of an Object Traveling in Circular Motion The average speed of an object is the distance traveled divided by the elapsed time. For motion along a circle, we distinguish between linear speed and angular speed . Finding the Area of a Sector of a Circle Find the area of the sector of a circle of radius 2 feet formed by an angle of 30 .° Round the answer to two decimal places. Solution Use formula (8) with r 2feet = and 30 6 θ π = ° = radian. [Remember, in formula (8), θ must be in radians.] A r 1 2 1 2 2 6 3 1.05 2 2 θ π π = = ⋅ ⋅ = ≈ The area A of the sector is 1.05 square feet, rounded to two decimal places. Now Work PROBLEM 79 EXAMPLE 7 Figure 18 = v s t u s r Time t DEFINITION Linear Speed Suppose that an object moves on a circle of radius r at a constant speed. If s is the distance traveled in time t on this circle, then the linear speed v of the object is defined as v s t = (9) As this object travels on the circle, suppose that θ (measured in radians) is the central angle swept out in time t. See Figure 18.
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