Section 6.1 Angles and Their Measure

Transcription

1 Section 6. Angles and Thei Measue A. How to convet Degees into Radians. To convet degees into adians we use the fomula below. Radians degees 80 Eample : Convet 0 0 into adians Radians degees adians Eample : Convet 0 0 into adians Radians degees adians Eample : Convet 50 0 into adians Radians degees adians Eample : Convet 0 into adians (Rounded of to decimal places) Radians degees (0.075) 0.0 adians Note: This means that 0 is the same angle as appoimatel 0.0 adians so ou can use this fact to help ou visualize what othe angles in adians will look like. Also the following angles ae useful ones to memoize is the same as adians is the same as adians 70 0 is the same as adians 60 0 is the same as adians Eample 5: Convet 7 0 into adians (Rounded of to decimal places) Radians degees 7 7 (0.075).7 adians Eample 6: Convet 0 0 into adians (Rounded of to decimal places) Radians degees 0 0 (0.075) 5. adians 80 80

2 B. How to convet Radians into Degees. To convet adians into degees we use the fomula below. Degees Radians 80 Eample : Convet adians into Degees Degees Radians Eample : Convet 5 adians into Degees 6 Degees Radians Eample : Convet 7 adians into Degees Degees Radians Eample : Convet adian into Degees (Rounded of to decimal places) Degees Radians (57.0) Eample 5: Convet.5 adian into Degees (Rounded of to decimal places) Degees Radians (57.0) Eample 6: Convet.5 adian into Degees (Rounded of to decimal places) Degees Radians (57.0).66 0

3 C. How to find the length of an ac. To find the length of an ac S, fo an angle θ in adians we use the fomula. s θ Eample : Find the length of the ac when θ and 5 inches s θ 5 5 inches The ac length S 5 inches o (.9 inches) Eample : Find the length of the ac when θ 0 0 and miles We must fist convet 0 0 into adians and then we can use the fomula. Radians degees adians s θ miles o (87.95 miles) The ac length S miles Eample : Find the angle θ when the ac length S ft and 0 ft s θ 0θ θ 0 0. θ The angle θ is 0. adians Eample : Find the adius of the ac when θ s θ and S 7 cm. cm The adius. cm

4 D. How to find the aea of a secto. To find the aea of a Secto A, fo an angle θ in adians we use the fomula. A θ Eample : Find the Aea of the secto when θ and 5 inches A θ A 5 5 A inches 8 The aea of the secto A 5 squae inches o 9.8 squae inches. Eample : Find the Aea of the secto when θ adians and miles A θ 8 A A sq miles The aea of the secto A squae miles. Eample : Find the angle θ when Aea of the secto A 5 sq cm and.5 cm A θ 5 (.5) θ 5.5θ 5 θ.5.8 θ The angle θ.8 adians Eample : Find the adius, when the angle θ adians and the Aea of the secto A sq ft A θ 7 7 The adius 7 ft o 8.9 ft

5 E. Applications. #9. Find the angle fo 5 minutes and convet to adians θ You can then use the fomula fo ac length s θ θ adians #9. Convet 5 0 to adians θ 5 adians 80 You can then use the fomula fo the aea of a secto A #95. Convet 5 0 to adians θ 5 adians 80 You can then use the fomula fo the aea of a secto A θ θ Distance #97. Linea Speed Time We know the Time 0 seconds The Distance is the length of an ac with a adius 5 cm and an angle θ adian #99. Distance tavelled b one evolution of the wheel C D (6) 8.67 inches 6.8 ft Speed 5 miles pe h 0.58 miles pe min580(0.58) ft pe min 078 ft pe min Numbe of evolutions Distance Speed Cicumfeence evolutions (appoimatel) 05. Linea Speed Time We know the time Time 7. das 655. hous The distance tavelled is one evolution of a cicle with adius We use the fact that the Distance is C 9,000 miles 09. Wheel does 0 evolutions pe minute, which will become 600 evolutions pe hou So its speed is its distance tavelled in hou which is 600 evolutions of a cicle with adius of ft. One evolution () 8 5. ft 600 evolutions 600(5. ft) 5,078 ft in an hou Speed 5,078 ft pe hou.86 miles pe hou. Radius 559 miles Distance moon tavels evolution (559),59 miles

6 Section 6. Tig Ratios A. Find eact values fo the si Tig Ratios fom the Unit Cicle. If the angle is between 0 o and 50 0 o between o adians and adians then we can use the Unit Cicle o the Table of Common Tig Ratios given on the pevious pages, Eample : Find the si tig atios fo the point (, ) sin tan cot csc sec Eample : Find the cos adians using unit cicle (o table) Eample : Find the sin adians this angle is lage than and so we cannot use the Unit cicle until we find an equivalent angle in the ange 0 to ange. We do this b subtacting fom. So an equivalent 6 5 So the angle sin Eample : Find the tan ( 60 o ) is equuivalent sin 5 5 adians using unit cicle (o table) 60 o this angle is less than 0 0 and so we cannot use the Unit cicle until we find an equivalent angle in the ange 0 to We do this b adding 60 0 to 60 o. So an equivalent 60 o using unit cicle (o table) tan

7 Eample : Find the sin (5 o ) + cos(5 0 ) B. Find eact values epessions. sin (5 o ) + cos(5 0 ) + Eample : Find the sec ( ) + cot( ) sec ( ) sec ( ) cot( ) cot( ) sec ( ) + cot( ) + Eample : Find the sin (5 o ) + sin(5 0 ) + sin (5 o ) + sin(5 0 ) sin (5 o ) sin (5 o ) sin (5 o ) sin (5 o ) sin (5 o ) + sin(5 0 ) + sin (5 o ) + sin(5 0 ) + 0 Eample : If θ 60 0 and f(θ) sin(θ) and g(θ) cos θ Find the eact vale of f(θ), g(θ), (f(θ)) g( θ ) f(θ) f(80 0 ) sin(80 0 ) 0 g(θ) g(60 0 ) cos(60 0 ) (f(θ)) ( f (60 0 )) ( sin (60 0 )) ( ) g( θ ) g(60 ) g(00 ) cos(0 0 )

8 C. Using ou Calculato to find appoimate values of Tig Ratios. You TI 8 and TI 8 calculatos have the abilit to find sin, cos and tangent atios ou must howeve make sue that ou calculato is in Degee Mode when using Degees and Radian Mode when using Radians. To change the mode pess the MODE ke on ou calculato and change the mode thee. If ou need to find the othe thee Tig Ratios Sec, cosec and cot then ou can use the following Tig popeties, cosec θ sinθ sec θ cosθ cotan θ tanθ Eample : Use ou calculato to find the following tig atios to decimal places (a) sin( 0 ) (b) cos(.) (c) tan(. 0 ) (d) csec( 5 ) (e) sec( 0 ) (f) cot(5.) (a) sin( 0 ) 0.07 (b) cos(.) (c) tan(. 0 ) 0.0 (d) csec( 5 ) csec(0.68) sin(0.68).70 (e) sec( 0 ) (f) cot(5.) cos( 0 ) tan(5.)

9 D. Find eact values fo the si Tig Ratios fom a Cicle whose adius is not. To solve these questions we need to use a evised fomula fo the si Tig Ratios when the cicle has a adius of. The si Tig Ratios ae given below. sin tan csc sec cot Eample : Find the si tig atios fo the following points. (a) (6,8) (b) (5, ) (c) (, ) Solution (a): Fo the point (6,0) the adius will be found b using pthagoas sin csc sec tan 8 6 cot 6 8 Solution (b): Fo the point (5, ) the adius will be found b using pthagoas. 5 + ( ) sin csc 5 sec 5 tan 5 5 cot 5 5 Solution (c): Fo the point (, )the adius will be found b using pthagoas. ( ) + ( ) + sin csc sec tan cot