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Figure 3.2 shows some special angles in standard position with the indicated terminal sides. The degree and radian measures are also given. "E. Ta nibs I 90% = 7 120 = 4 135 = 150 = 6 30%= 180 = 7 0' = 0; 360 = 27 2100 = - 7 T 3300 - lin 6 2250 = 571 3150 = 4 3009= 240 = 3 3T Example 1. Express 750 and 240" in radians. Solution. 75 750 5TT 180 12 12 rad 240 47T 180 240 4 7 3 rad 3 Example 2. Express - rad and - rad in degrees. Solution. 180 00 1 7 TT = 22.5 rad = 22.50 117 / 180 117 = 330 6 6 rad = 330 Coterminal Angles Two angles in standard position that have a common terminal side are called coterminal angles, Observe that the degree measures of coterminal angles differ by multiples of 369 singon busbnsia Two angles are coterminal if and only if their degree measures differ by 360k, where ke Z. lawn ,unibri of gotgob gui liano ) Similarly, two angles are coterminal if and only if their radian mea- sures differ by 27k, where ke Z.JaRain 951895 4 ITWAS alExample 1. Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30. Solution. Since the given central angle is in degrees, we have to convert it into radian measure. Then apply the formula for an arc length. 30 180 rad 10 3 m Example 2. A central angle 0 in a circle of radius 4'm is subtended by an arc of length 6 m. Find the measure of 0 in radians. Solution. rad de- hiw lenimmstop signs sch bond I sigman A sector of a circle is the portion of the interior of a circle bounded by the bad is initial and terminal sides of a central angle and its intercepted arc. It is like a isd (d "slice of pizza." Note that an angle with measure 27 radians will define a sector that corresponds to the whole "pizza."b Therefore, if a central angle of a sector nostirio? has measure 0 radians, then the sector makes up the fraction , of a complete O ni asif circle. See Figure 3.5. Since the area of a complete circle with radius r is ur?, we have PONE Ud8 5 1 2088 (1 ) Area of a sector = 2T (TT? ) = -or2. 09 - 028 + 088- (9) tonio linn & ni lyas A 20 Abel peodw Figure 3.5 In a circle of radius r, the area A of a sector with a central angle measuring 0 radians is A = 20. Example 1. Find the area of a sector of a circle with central angle 60" if the radius of the circle is 3 mewines s vd bigos stat is as lo e aignol on a dubai to offers a nlbatseibai and dew noisizog batonsle ni signs labege smos aworda $.& su.i'd 1 radian ~ 57.30 Re * #3 081 Examples. In the following figure, identify the terminal side of an angle in standard position with given measure. (1) degree measure: 135, -135, -90, 405 - COTS (2) radian measure: * rad, - " rad, " rad, - , rad 2nsibm ni De bas "25 cesiquil I siqmexil C B ( DH) ) AT 450 A X, 450 1350 D E 08 1 TO Solution. (1) 135: OC; -1359: OB; -90: OF; and 405: OB (2) radian measure: " rad: OB; -3" rad: OD; 3: rad: OF; and -" rad: OF 2sign/ Teaimsto.) LESSON Convert degree measure to radian measure, and vice versa and Illustrate angles in standard position and coterminal angles. Converting degree to radian, and vice versa , 1098 vd som nBiber 191 1 li ino bus fi stimetos onebigus ow yhelimia 1. To convert a degree measure to radian, multiply it by 180.lib gama 2. To convert a radian measure to degree, multiply it by 180.* As a quick illustration, to hild ine coterinnal angle with an angle that beef women sures 410, just subtract 360, resulting in 50. See Figure 3.3. 50okay watered by the frankle bri ansiber ni 0 to swanson sell Example 1. Find the angle coterminal with -380 that has measure a) between Ofand 360%d glorio s lo niemi ard to noitog ads zi alone a to 1040se A b) between -360 and O'is bolquisini ali bats signs laTing, a to 29bie Isainnes bus Initial Tojose s onfeb lliw ansibal TS suresent dew signs as feds ston ".saxiq To goile" Solution. A negative angle moves in a clockwise direction, and the angleba3809noo sad lies in Quadrant IV. to _ notJosi eds qu 29xsi 101592 91 add ausiben 0 9TuaRom ead (1) -380 + 2 . 360 = 340 9ved (2) -380 + 360 = -20 To1398 8 10 891A O LESSON Illustrate the relationship between the linear and angular measures of a central angle in a unit circle. Arc Length and Area of a Sector In a circle, a central angle whose radian measure is 0 subtends an arc that is the fraction of the circumference of the circle. Thus, in a circle of radius r (see Figure 3.4), the length s of an arc that subtends the angle 0 is S = x circumference of circle = 2 (2TT) = re. i BuBibsT 0 animaRom Figure 3.4 In a circle of radius r, the length s of an arc intercepted by a central angle with measure 0 radians is given by s = re.Angle An angle is formed by rotating a ray about its endpoint. In the figure shown below, the initial side of ZAOB is OA, while its terminal side is OB. An angle is said to be positive if the ray rotates in a counterclockwise direction, and the angle is negative if it rotates in a clockwise direction. 2101)12181 :sour. Initial side A terminal side negative angle positive angle terminal side Initial side Unit Circle ol A unit circle is a circle whose radius has a length of one unit, [ = 1 (0, 1) bisb (, 0) LE VACI An angle is in standard position if it is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis. The angles a, B and 0 in the following figure are angles in standard position. byinn't A 28-08 gg b.14- B Angle measure is the amount of the rotation done by the terminal side away from the initial side. A 10 angle is equal to 1/360 of a complete revolution of the terminal side, 7. will old Ded TATTOO 1 revolution = 360 Radian Measure A radian (1 rad) is the measure of the central angle subtended by an arc of a circle whose length is equal to the radius of the circle. nonoubeitel' 6(m)=rad 1 1r 3?: A=- 2________ 2 2(3)3 2m ' is set to spray water over a distance of 70 feet fairway watered by the sprinkler. Solution 1r 2,; 21r __ 490M 5131 g Activity 1: Identify the terminal side of an angle in standard position with given measure. 1. degree measure: 60; -210 2. radian measure: - 3 to find the exact value of sesh expre tan( 2 Activity 2 A. 1. Convert the following degree measures to radian measure. (a) 60 (b) 90 2. Convert the following radian measures to degree measure. (a) grad (b) 37 rad B 1. Find the angle between 0 and 360 (if in degrees) or between 0 rad and 27 rad (if in radians) that is coterminal with the given angle. (a) 7360 (b) -2848'65" (c) 131 rad 2 (d) 10 rad