Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic tigonometic functions in tems of a ight tiangle and the measues of its thee sides. Befoe beginning ou study of tigonomety, we need to take a look at some basic concepts having to do with angles. An angle is fomed by two ays that shae a common endpoint, called the vetex of the angle. One ay is called initial side of the angle, and the othe side is called the teminal side. Fo ease, we typically will daw angles in the coodinate plane with the initial side along the positive x axis. We measue angles in two diffeent ways, both of which ely on the idea of a complete evolution in a cicle. You ae pobably familia with degee measue. In this system of angle measue, an 1 th angle which is one complete evolution is 360. So one degee is of a cicle. 360 1
The second method is called adian measue. One complete evolution is. Suppose I daw a cicle and constuct an angle by dawing ays fom the cente of the cicle to two diffeent points on the cicle in such a way that the length of the ac intecepted by the two ays is the same as the adius of the cicle. The measue of the cental angle thus fomed is one adian. θ = 1 ac length = Radian measue of an angle: Place the vetex of the angle at the cente of a cicle of adius. Let s denote the length of the ac intecepted by the angle. The adian measue θ of the angle is the atio of the ac s length s to the adius. That is, θ =. In geneal, the adian measue of a cental angle θ can be detemined by the fomula s θ =, whee s is the length of the intecepted ac and is the adius of the cicle and and s ae measued in the same units.
Example 1: A cicle has adius 1 inches. A cental angle θ intecepts an ac of length 36 inches. What is the adian measue of θ? We know that the cicumfeence of a cicle is. In this case, θ = =. So the adian measue of the cental angle in the case of a complete evolution is. Compaing the two systems, then, we have that adians = 360 adians = 180 adians = 90 etc. As you ae becoming moe familia with adian measue, you may find it helpful to be able to convet between the two systems. We can use the statement adians = 180 to help do this. Dividing both sides of that equation by, we have that 1 adian = 180 180 so to convet to degees, multiply by. Similaly, 1 degee = 180 so to convet to adians, multiply by 180. These ae the convesion fomulas fo adians to degees and fo degees to adians, espectively. 3
1 adian = 180 180 so to convet to degees, multiply by. 1 degee = 180 so to convet to adians, multiply by 180. Example : Convet 135 to adian measue. Example 3: Convet 4 to degees. 3 Example 4: Convet to degees. 9 Example 5: Convet 18 to adian measue. You will use some angles so often that you should know both thei degee and adian measues. These ae: 30 = 6 45 = 4 60 = 3 90 = 180 = 360 = Memoize these! 4
If s θ =, then we can multiply both sides of this equation by, which gives us s = θ. This is called the aclength fomula and it gives the length of the ac intecepted by the cental angle. Note, to use this fomula the angle measue MUST be given in adians. Example 6: If the adius of a cicle is 16 inches and the measue of its cental angle is 3, find the aclength of the secto intecepted by the angle. 4 Example 7: If the aclength of a secto is 8 cm. and the adius is 1 cm., find the measue of the cental angle. 5
A secto of a cicle is the egion bounded by a cental angle and the intecepted ac. Sometimes, you ll need to find the aea of a secto. The fomula fo the aea of a cicle is A=. A secto is a faction of a cicle, detemined by the measue of its cental angle ove the complete evolution that is a cicle, that is. So the aea of a section is θ this faction of the aea of the cicle, that is: θ θ 1 A= = = θ. Note, to use this fomula, the measue of the cental angle must be given in adians. Example 8: A secto has adius 10 and cental angle measuing.5 adians. Find the aea of the secto. 6
Example 9: A secto has cental angle measuing 5 adians. The aea of the secto is 500 squae units. Find the adius. Example 10: Find the peimete of a secto with cental angle 60 and adius 3 m. Example 11: If the aea of a secto is m and the measue of the cental angle is 4, find the adius. 7
Angula and Linea Velocity Suppose you ae iding on a mey-go-ound. The ide tavels in a cicula motion, and the hoses usually move up and down. Some of the hoses ae ight along the edge of the mey-go-ound, and some ae close to the cente. If you ae on one of the hoses at the edge, you will tavel fathe than someone who is on a hose nea the cente. But the length of time that both people will be on the ide is the same. If you wee on the edge, not only did you tavel fathe, you also taveled faste. Howeve, eveyone on the mey-go-ound tavels though the same numbe of degees (o adians). Thee ae two quantities we can measue fom this, angula velocity and linea velocity. The angula velocity of a point on a otating object is the numbe of degees (o adians o evolutions) pe unit of time though with the point tuns. This will be the same fo all points on the otating object. The linea velocity of a point on the otating object is the distance pe unit of time that the point tavels along its cicula path. This distance will depend on how fa the point is fom the axis of otation (the cente of the mey-go-ound). We let the Geek lette ω epesent angula velocity. Using the definition above, θ ω = t We denote linea velocity by v. Using the definition above, aclength. The elationship between these two quantities is given by adius. v = a t v = ω, whee a is the, whee is the 8
θ ω = t a v = v = ω t Example 1: If the speed of a evolving gea is 5 pm, a. Find the numbe of degees pe minute though which the gea tuns. b. Find the numbe of adians pe minute though which the gea tuns. Example 13: A ca has wheels with a 10 inch adius. If each wheel s ate of tun is 4 evolutions pe second, a. Find the angula speed in units of adians/second. b. How fast (linea speed) is the ca moving in units of inches/second? 9
Example 14: A CD spins at the ate of 500 evolutions pe minute. How many degees pe minute is this? 10