CHAPTER 4 Trigonometry

Transcription

1 CHAPTER Trigonometr Section. Radian and Degree Measure You should know the following basic facts about angles, their measurement, and their applications. Tpes of Angles: (a) Acute: Measure between 0 and 90. Right: Measure 90. (c) Obtuse: Measure between 90 and 80. (d) Straight: Measure 80. and are complementar if The are supplementar if Two angles in standard position that have the same terminal side are called coterminal angles. To convert degrees to radians, use 80 radians. To convert radians to degrees, use radian 80. one minute 0 of. one second 0 of of. The length of a circular arc is s r where is measured in radians. arc length Linear speed s time t Angular speed t srt Vocabular Check. Trigonometr. angle. coterminal. radian 5. acute; obtuse. complementar; supplementar 7. degree 8. linear 9. angular 0. A r... radians. 5.5 radians. radians radians. radian..5 radians. 5

2 Chapter Trigonometr 7. (a) Since 0 < lies in Quadrant I. 5 < ; 5 < 7 lies in Quadrant III. 5 < ; (a) Since < < lies in Quadrant III. 8 ; 8 < 9 lies in Quadrant III. 8 < ; (a) Since lies in Quadrant IV. < < 0 ; < < lies in Quadrant III. ; 0. (a) Since < < 0; lies in Quadrant IV. lies in Quadrant II. < <, 9 9. (a) Since <.5 < ;.5 lies in Quadrant III.. (a) Since lies in Quadrant IV. <.0 < ;.0 <.5 < ;.5 lies in Quadrant II. <.5 < ;.5 lies in Quadrant II.. (a) 5. (a) 7 5. (a) 5π π 7 π 5 π 5π. (a) 7 7π

3 Section. Radian and Degree Measure 7 7. (a) Coterminal angles for Coterminal angles for (a) (a) Coterminal angles for 8 Coterminal angles for 5 0. (a) (a) Complement: Complement: Not possible, is greater than.. (a) Complement: Complement: Not possible, 5 is greater than.. (a) Complement: Complement: Not possible, is greater than (a) Complement: Not possible, is greater than. 5. Complement: º º. 0.

4 8 Chapter Trigonometr (a) Since 90 < 0 < 80, 0 lies in Quadrant II.. (a) Since 0 < 8. < 90, 8. lies in Quadrant I. 70 < 85 < 0, 85 lies in Quadrant IV. 80 < 57 0 < 70, 57 0 lies in Quadrant III.. (a) Since 80 < 50 < 90, 50 lies in Quadrant III. 0 < < 70, lies in Quadrant I.. (a) Since 70 < 0 < 80, 0 lies in Quadrant II. 90 <. < 0,. lies in Quadrant IV. 5. (a) 0. (a) (a) (a)

5 Section. Radian and Degree Measure 9 9. (a) Coterminal angles for 5 0. (a) (a) Coterminal angles for Coterminal angles for Coterminal angles for (a) (a) Complement: Complement: Not possible, 5 is greater than (a) Complement: Complement: (a) Complement: Complement: Not possible, 50 is greater than (a) Complement: Not possible, 0 is greater than Complement: Not possible, 70 is greater than (a) (a) (a) (a) (a) (a) (a) (a) radians radians radians radians radians radians 80

6 0 Chapter Trigonometr radian radians (a) (a) (a) (a) (a) (a) (a) (a) s r 5 5 radians 80. s r radians 8. s r radians 8. s r 8. s r 8. r feet, s 8 feet radians Because the angle represented is clockwise, this angle is radians radians s r 8 7 radians

7 Section. Radian and Degree Measure 85. s r radians 8. r 80 kilometers, s 0 kilometers s r 0 80 radians 87. s r, in radians s inches 7. inches r 9 feet, 89. s r, in radians s r 9 s meters feet 9. feet 90. r 0 centimeters, s r 0 5 centimeters 5.7 centimeters 9. A r 9. r mm, 9. A r A 8 square inches A r A square inches 8 mm.7 square feet 5.55 mm 9. r. miles, A square miles radian s r miles 9. r 000 miles s r radian radian s r miles 98. r 89 kilometers 99. s r The difference in latitude is about 0.07 radians.59. s r radian 00. s r 5.8 radians (a) 5 miles per hour feet per minute 0 The circumference of the tire is C.5 feet. The number of revolutions per minute is r revolutions per minute.5 The angular speed is t Angular speed 57 radians 57 radians minute 57 radians per minute

8 Chapter Trigonometr 0. Linear velocit for either pulle: inches per minute (a) Angular speed of motor pulle: Angular speed of the saw arbor: v r v r Revolutions per minute of the saw arbor: radians per minute radians per minute revolutions per minute 0. (a) Angular speed 500 radians minute 0,00 radians per minute,7.5 radians per minute Linear speed 7.5 in. ft in. 500 minute 0. (a) rpm radians per minute 8 radians per minute 5. radians per minute r 5 ft r t 00 feet per minute feet per minute feet per minute Linear speed 55.7 feet per minute 8. feet per minute 05. (a) 00 Angular speed 500 radians per minute Interval: 00, 000 radians per minute 00 Linear speed 500 centimeters per minute Interval: 00, 000 centimeters per minute 0. A R r 07. A r R 5 5 r 5 A r square inches square meters 9. square meters (a) Arc length of larger sprocket in feet: s r Therefore, the chain moves feet, as does the smaller rear sprocket. Thus, the angle of the smaller sprocket is r inches feet. s r feet feet seconds CONTINUED feet 00 seconds hour and the arc length of the tire in feet is: s feet s r s feet Speed s t second mile 0 miles per hour 580 feet feet per second

9 Section. Radian and Degree Measure 08. CONTINUED the arc length of the tire is feet and the cclist is pedaling at a rate of one revolution per second, we have: (c) Distance Distance Rate Time feet per second mile 7 t seconds 580 feet 790 t miles (d) The functions are both linear. feet revolutions mile 7 n revolutions 580 feet 790 n miles 09. False. An angle measure of radians corresponds to two complete revolutions from the initial to the terminal side of an angle. n0 and 0. True. If and are coterminal angles, then where n is an integer. The difference between is n0 n.. False. The terminal side of 0 lies on the negative -ais.. (a) An angle is in standard position if its verte is at the origin and its initial side is on the positive -ais. A negative angle is generated b a clockwise rotation of the terminal side. (c) Two angles in standard position with the same terminal sides are coterminal. (d) An obtuse angle measures between 90 and 80.. Increases, since the linear speed is proportional to the radius. 80. radian 57., so one radian is much larger than one degree. 5. The arc length is increasing. In order for the angle to remain constant as the radius r increases, the arc length s must increase in proportion to r, as can be seen from the formula s r.. The area of a circle is A r A The circumference of a circle is C r. r. C A r r C A r Cr A For a sector, C s r. Thus, A rr for a sector. r

10 Chapter Trigonometr. f 5. Graph of 5 shifted to the right b two units = 5 f 5 Vertical shift four units downward = 5 = 5 = ( ) 5. f 5. f 5 = ( + ) 5 Graph of 5 reflected in -ais and shifted upward b two units 5 = 5 Reflection in the -ais and a horizontal shift three units to the left = 5 = 5 5 Section. Trigonometric Functions: The Unit Circle You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and the point on the unit circle corresponding to t., sin t csc t, 0 cos t sec t, 0 tan t, 0 cot t, 0 The cosine and secant functions are even. cost cos t sect sec t The other four trigonometric functions are odd. sint sin t csct csc t tant tan t cott cot t Be able to evaluate the trigonometric functions with a calculator. Vocabular Check. unit circle. periodic. period. odd; even