4.1 Radian and Degree Measure


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1 4. Radian and Degree Meaure An angle AOB (notation: AOB ) conit of two ray R and R with a common vertex O (ee Figure below). We often interpret an angle a a rotation of the ray R onto R. In thi cae, R i called the initial ide of the angle, and R i called the terminal ide of the angle. If the rotation i counterclockwie, the angle i conidered poitive, and if the rotation i clockwie, the angle i conidered negative. The meaure of an angle i the amount of rotation about the vertex required to move R onto R. Intuitively, thi i how much the angle open. One unit of meaurement for angle i the degree. An angle of meaure degree i formed by rotating the initial ide of a complete revolution. In 360 calculu and other branche of mathematic, a more natural method of meauring angle i ued radian (abbreviated rad) meaure. Definition of Radian: one radian i the meaure of a central angle that intercept an arc equal in length to the radiu r of the circle. Algebraically, thi mean that where i meaured in radian. r
2 The circumference of the circle of radiu r i r, it follow that a central angle of one full revolution (counterclockwie) correpond to an arc length of r. Becaue the radian meaure of an angle of one full revolution i, you can obtain the following. revolution= radian, revolution= radian, revolution= radian Thee and other common angle are hown in the following. Recall that the four quadrant in a coordinate ytem are numbered I,II, III, and IV. The following graph how which angle between 0 and lie in each of the four quadrant. Note that angel between 0 and are called acute angle and angle between and are obtue angle. Two angle are coterminal if they have the ame initial and terminal ide. A given angle ha infinitely many coterminal angle: n, where n i an integer.
3 Two poitive angle and are complementary (complement of each other) if. Two poitive angle Example: and are upplementary (upplement of each other) if. a. 9, are coterminal? (Ye) 3 3 b. and are complementary? (ye) 3 6 c. 4 and are upplementary? (ye) 5 5 d. If poible, find the complement of 5. (no complement) 8 The econd way to meaure angle i in term of degree, denoted by the ymbol. A meaure of one degree ( ) i equivalent to a rotation of of a complete revolution about the vertex. To meaure 360 angle, it i convenient to mark degree on the circumference of a circle. A full revolution (counterclockwie) correpond to 360, a half revolution to 80. Converion between degree and radian. rad a. To convert degree to radian, multiply degree by b. To convert radian to degree, multiply radian by rad Example: Converting degree to radian or radian to degree. a. 30? 6 b. 45? 4 c.? 90? 80 d. e.? 35 f ?
4 The radian meaure formula,, can be ued to meaure arc length given by r r Arc Length. For a circle of radiu r, a central angle intercept an arc of length given by r where i meaured in radian. Example: A circle ha a radiu of inche. Find the length of the arc intercepted by a central angle of 40. (anwer: 8 ) 3 Linear and Angular Speed. Conider a particle moving at a contant peed along a circular arc of radiu r. If i the length of the arc traveled in time t, then the linear peed v of the particle i Linear peed arc length v time t Moreover, if i the angle (in radian meaure) correponding to the arc length, then the angular peed of the particle i central angle Angular peed time t Example: The econd hand of a clock i 0. centimeter long, a hown in the following graph. Find the linear peed of the tip of thi econd hand a it pae around the clock face. r (0. centimeter) Solution: Linear peed.068 centimeter per econd t 60 econd 60 econd Example. The blade of a wind turbine are 6 feet long. The propeller rotate at 5 revolution per minute. a. Find the angular peed of the propeller in radian per minute. b. Find the linear peed of the tip of the blade.
5 30 radian Solution: Angular peed 30 radian per minute t minute r 6 30 feet Linear peed 0,933 feet per minute t t minute A ector of a circle i the region bounded by two radii of the circle and their intercepted arc. Area of a Sector of a circle. For a circle of radiu r, the area A of a ector of the circle with central angle i given by A r Example: A prinkler on a golf coure fairway pray water over a ditance of 70 feet and rotate through an angle of 0. Find the area of the fairway watered by the prinkler. Solution: 0 radian 3 A quare feet 3 r
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