Topic/Objective: Essential Question: How do solve problems involving radian and/or degree measure?
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1 Topic/Objective: 4- RADIAN AND DEGREE MEASURE Name: Class/Peiod: Date: Essential Question: How do solve poblems involving adian and/o degee measue? Questions: TRIGONOMETRY. Tigonomety, as deived fom the Geek language, means measuement of angles. ANGLE. An angle is detemined by otating a ay about its endpoint. INITIAL SIDE. The initial side is the stating line of fomation of an angle. VERTEX. The vetex is the endpoint of the ay. Angles ae named fo thei vetices and ae denoted by the Geek lettes Alpha, Beta, and Theta, as well as A, B, and C. ANGLES IN THE COORDINATE PLANE VERTEX. The vetex of an angle in the coodinate plane is the oigin. STANDARD POSITION. An angle in the coodinate plane with its vetex at the oigin is said to be in standad position. VERTE X TERMIN AL INITIA L TERMIN AL INITIAL SIDE TYPES OF ANGLES POSITIVE ANGLES. Positive angles ae geneated by a counteclockwise otation. In the diagam, α is a positive angle. NEGATIVE ANGLE. Negative angles ae geneated by a clockwise otation. In the diagam, β is a negative angle.
2 RADIAN. Let be the measue of the cental angle of a cicle and s be the mino ac fomed by the cental angle. Then, one adian is s the measue of cental angle that intecepts an ac s equal in length to the adius of the cicle. Since the cicumfeence of a cicle is, the cental angle of one full counteclockwise evolution coesponds to an ac length of s. Thus, adians is equal to 60. In geneal, the measue of the cental angle is found using the fomula s s. CONVERT REVOLUTIONS TO RADIANS. To convet evolutions to adians, use the following pocedues.. Use the fact that one evolution equals.. Find the poduct of the numbe of evolutions and.. Reduce the esulting faction. EXAMPLE. Convet the following evolutions to adians. a. 4 b. 6 c. d. 8 e. f. 7 RADIAN MEASURE IN THE COORDINATE PLANE QUADRANT I. Positive angles of : 0. Negative angles of :. QUADRANT II. Positive angles of :. Negative angles of :.
3 QUADRANT III. Positive angles of :. Negative angles of :. QUADRANT IV. Positive angles of :. Negative angles of : 0. DETERMINE THE QUADRANT IN WHICH THE TERMINAL SIDE OF AN ANGLE LIES. To detemine the Quadant in which the teminal side of an angle, measued in adians, lies, use the following pocedues.. Detemine the measue of the teminal angle. a. If, state the Quadant in which the teminal side lies. b. If, subtact n (multiples of one evolution), whee n is an intege, so that. Then state the Quadant in which the teminal side lies. c. If, add n (multiples of one evolution), whee n is an intege, so that. Then state the Quadant in which the teminal side lies. EXAMPLE. State in which Quadant o along which axis the teminal side of each angle lies. a. b. 6 4 c. d. COMPLEMENTARY AND SUPPLEMENTARY ANGLES. Only positive angles can be complementay o supplementay. COMPLEMENTARY ANGLES. Two angles, measued in adians, ae complementay if the sum of thei measues equal. SUPPLEMENTARY ANGLES. Two angles, measued in adians, ae supplementay if the sum of thei measues equal. EXAMPLE. Detemine the complement and supplement of each angle if it exists. a. b. 4 5 DEGREE MEASURE. Degee measue is an angula measuement in which one degee equals of a evolution aound the cicle. 60
4 CONVERTING BETWEEN DEGREES AND RADIANS. To convet between degees and adians, use the following pocedues. DEGREES TO RADIANS. To convet degees to adians, multiply by 80. RADIANS TO DEGRESS. To convet adians to degees, multiply by 80. EXAMPLE 4. Expess each angle in degees o adian measue. Round the degee measue to the neaest hundedth if necessay. a. 5 b. 70 c. d. ad ARC LENGTH. The definition of adian measue, length of an ac of a cicle. s can be used to find the EXAMPLE 5. a. A cicle has a adius of 4 inches. Find the length of the ac, in adians, intecepted by a cental angle of 40 b. Given A, if m DAE = 85 and AD 4 detemine m DQE in adians. LINEAR AND ANGULAR SPEED. Conside a paticle moving at a constant speed along a cicula ac of adius. If s is the length of the ac taveled in time t, then ac length s the linea speed of the paticle is linea speed. time t If is the angle (in adian measue) coesponding to the ac length s, the angula cental angle speed of the paticle is angula speed. time t EXAMPLE 6. a. The second hand of a clock is 0. centimetes long. ) Find the linea speed of the tip of the second hand. ) Find the angula speed of the tip of the second hand. 4
5 b. A 0-inch adius lawn olle makes. evolutions pe second. ) Find the angula speed of the olle in adians pe second. ) Find the speed of the tacto that is pulling the olle. SUMMARY: 5
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