Radian and Degree Measure

Transcription

1 CHAT Pe-Calculus Radian and Degee Measue *Tigonomety comes fom the Geek wod meaning measuement of tiangles. It pimaily dealt with angles and tiangles as it petained to navigation, astonomy, and suveying. Today, the use has expanded to involve otations, obits, waves, vibations, etc. Definitions: An angle is detemined by otating a ay (half-line) about its endpoint. The initial side of an angle is the stating position of the otated ay in the fomation of an angle. The teminal side of an angle is the position of the ay afte the otation when an angle is fomed. The vetex of an angle is the endpoint of the ay used in the fomation of an angle. Teminal side Vetex Initial side 1

2 CHAT Pe-Calculus An angle is in standad position when the angle s vetex is at the oigin of a coodinate system and its initial side coincides with the positive x-axis. y Teminal side Initial side x A positive angle is geneated by a counteclockwise otation; wheeas a negative angle is geneated by a clockwise otation. y y Positive angle x x Negative angle

3 CHAT Pe-Calculus If two angles ae coteminal, then they have the same initial side and the same teminal side. β α Radian Measue Definitions: The measue of an angle is detemined by the amount of otation fom the initial side to the teminal side. A cental angle is one whose vetex is the cente of a cicle. One adian is the measue of a cental angle Ө that intecepts an ac s equal in length to the adius of the cicle.

4 CHAT Pe-Calculus s=10 θ = 10 θ is 1 adian in size. How many adians ae in a cicle? θ is 1 adian θ about 1 of θ about 1 of a adian Thee ae about 1 adians in a cicle.

5 β = 10 s=15 If s = 15, then we need to find out how many s ae in s to know the numbe of adians in β. CHAT Pe-Calculus 15 = 1.5, so β is 1.5 adians. 10 *In geneal, the adian measue of a cental angle θ with adius and ac length s is θ = s We know that the cicumfeence of a cicle is π. If we conside the ac s as being the cicumfeence, we get θ = = π This means that the cicle itself contains an angle of otation of π adians. Since π is appoximately.8, this matches what we found above. Thee ae a little moe than adians in a cicle. (π to be exact.) Theefoe: A cicle contains π adians. A semi-cicle contains π adians of otation. A quate of a cicle (which is a ight angle) contains π adians of otation. 5

6 CHAT Pe-Calculus Definition: A degee is a unit of angle measue that is equivalent to the otation in 1/0 th of a cicle. Because thee ae 0 in a cicle, and we now know that thee ae also π adians in a cicle, then π = 0. 0 = π adians π adians = = π adians 1π adians = = adians adian = To convet adians to degees, multiply by 180. To convet degees to adians, multiply by. 180

7 CHAT Pe-Calculus Example: Convet 10 to adians. 10 = 10( ) = 10 = Example: Convet -15 to adians. -15 = -15( ) = 15 = Example: Convet 5 to degees Example: Convet 7 to degees This makes sense, because 7 adians would be a little moe than a complete cicle, and is a little moe that 0 *Notice: If thee is no unit specified, it is assumed to be adians. 7

8 CHAT Pe-Calculus 8 Degee and Radian Equivalent measues y x

9 Definition: Two positive angles α and β ae complementay if thei sum is o 90. CHAT Pe-Calculus Two positive angles α and β ae supplementay if thei sum is π o 180. Definition: An acute angle has a measue between 0 and (o between 0 and 90.) An obtuse angle has a measue between and π (o between 90 and 180.) Example: Find the supplement and complement of. 5 complement: 5 10 supplement: 5 5 9

10 CHAT Pe-Calculus Coteminal Angles Two angles ae coteminal if they have the same initial side and the same teminal side. Look at and is the same as with π added to it. *In geneal, you can find an angle that is coteminal with an angle θ by adding o subtacting multiples of π. (Each multiple of π is a full evolution aound the cicle.) We wite this as θ + kπ, whee k is an intege. 10

11 CHAT Pe-Calculus Examples of coteminal anlges: Example: Find an angle that is coteminal with θ = One possible solution: Example: Find an angle that is coteminal with θ =. One possible solution: 8 5. Example: List all of the angles that ae coteminal with answe: ± kπ 11

12 CHAT Pe-Calculus Calculato Convesion Factional pats of degees can be denoted as decimal degees o as degees, minutes and seconds. 1 = 0 (minutes) 1 = 0 (seconds) This also means that, 1 = = 1 o example: 15 0 *To convet to decimal degees: On you gaphing calculato: Ente Use the [ANGLE] menu fo and and [ALPHA] [+] fo the. Pess [MATH] [ Dec] [ENTER] to convet to decimal degees. 1

13 Example: Convet 8 to decimal degees. answe: 8.79 CHAT Pe-Calculus To convet decimal degees to degees, minutes, and seconds (DMS): Ente the decimal degee. Pess [ANLGE] [ DMS] [ENTER] to convet. Round the seconds to the neaest second. Example: Convet to degees, minutes and seconds. answe: Applications Because we aleady know that with adian measue θ = s, whee s is the ac length, then s = θ. Example: Find the length of the ac that subtends a cental angle with measue 10 in a cicle with adius 5 inches. Change to adians. s inches 180 1

14 CHAT Pe-Calculus Linea and Angula 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 the linea speed of the paticle is Linea speed = ac length time = s o t t Moeove, if θ is the angle (in adian measue) coesponding to the ac length s, then the angula speed of the paticle is Angula speed = cental angle time = t *Note: Linea speed can also be epesented as s o t s t t t (adius)(angula speed) Example: The cicula blade on a saw otates at 00 evolutions pe minute. (a) Find the angula speed in adians pe second. 1

15 CHAT Pe-Calculus Because each evolution geneates π adians, it follows that the saw tuns (00)(π) = 800 π adians pe minute. In othe wods, the angula speed is Angula speed t 800 adians 80 adianspe 0 seconds second (b) The blade has a diamete of 1 inches. Find the linea speed of a blade tip. s Linea speed = t t (8 inches) (80 ) (1 second) 010. inches/second If we use t=1, then fom pat (a) we know that θ is 80π because that is the adians in one second s otation. Altenately, Linea speed = t (adius)(angula speed) (8 inches) (80 adians/second) 010. inches/second 15