In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

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1 Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In many scientific and engineeing calculations adians ae used in pefeence to degees. In ode to maste the techniques explained hee it is vital that you undetake plenty of pactice execises so that they become second natue. Afte eading this text, and/o viewing the video tutoial on this topic, you should be able to: use adians to measue angles convet angles in adians to angles in degees and vice vesa find the length of an ac of a cicle find the aea of a secto of a cicle find the aea of a segment of a cicle Contents 1. Intoduction 2 2. Definition of a adian 2 3. Ac length 3 4. Equivalent angles in degees and adians 4 5. Finding an ac length when the angle is given in degees 5 6. The aea of a secto of a cicle 6 7. Miscellaneous examples 6 1 c mathcente June 11, 2004

2 1. Intoduction At school we usually lean to measue an angle in degees. otation is 360 as shown in Figue 1. We ae well awae that a full 360 o Figue 1. A full otation is 360. Howeve, thee ae othe ways of measuing an angle. One way that we ae going to have a look at hee is measuing angles in units called adians. In many scientific and engineeing calculations adians ae used in pefeence to degees. 2. Definition of a adian Conside a cicle of adius as shown in Figue 2. 1 ad Figue 2. The ac shown has a length chosen to equal the adius; the angle is then 1 adian. In Figue 2 we have highlighted pat of the cicumfeence of the cicle chosen to have the same length as the adius. The angle at the cente, so fomed, is 1 adian. Key Point An angle of one adian is subtended by an ac having the same length as the adius as shown in Figue 2. c mathcente June 11,

3 3. Ac length We will now use this definition to find a fomula fo the length of an abitay ac. We have seen that an angle of 1 adian is subtended by an ac of length as illustated in the left-most diagam in Figue 3. By extension an angle of 2 adians will be subtended by an ac of length 2, as shown Figue 3. An angle of 2 adians is subtended by an ac of length 2. Note fom these diagams that the length of the ac is always given by the angle in adians the adius In the geneal case, the length s, of an abitay ac which subtends an angle is as illustated in Figue 4. s Figue 4. The ac length s, is given by This gives us a way of calculating the ac length when we know the angle at the cente of the cicle and we know its adius. Key Point ac length s = (note: must be measued in adians) Execise 1 Detemine the angle (in adians) subtended at the cente of a cicle of adius 3cm by each of the following acs: a) ac of length 6 cm b) ac of length 3π cm c) ac of length 1.5 cm d) ac of length 6π cm 3 c mathcente June 11, 2004

4 4. Equivalent angles in degees and in adians We know that the ac length fo a full cicle is the same as its cicumfeence, 2π. We also know that the ac length =. So fo a full cicle 2π = that is =2π In othe wods, when we ae woking in adians, the angle in a full cicle is 2π adians, in othe wods 360 =2π adians This enables us to have a set of equivalences between degees and adians. fom which it follows that Key Point 360 =2π adians 180 = π adians 90 = π 2 adians 45 = π 4 adians 60 = π 3 adians 30 = π 6 adians The Key Point gives a list of angles measued in degees on the left and the equivalent list in adians on the ight. It is impotant in mathematical wok that you ecod coectly the unit of measue you ae using. Anothe useful elationship is given as follows: π adians = 180 so 1 adian = 180 π degees = (3 d.p.) So 1 adian is just ove 57. Some notation. Thee ae vaious conventions used to denote adians. Some books and some teaches use ads as in 2 ads. Othes use a small c as in 2 c. Some othes use no symbol at all and assume that adians ae being used. When an angle is expessed as a multiple of π, fo example as in the expession sin 3π, it is taken as ead that the angle is being measued in adians. 2 c mathcente June 11,

5 Execise 2 1. When each of the following angles is conveted fom degees to adians the answe can be expessed as a multiple of π (note that it may be a factional multiple). In each case state the multiple (e.g fo an answe of 4π 5 the multiple is 4 5 ). a) 90 o b) 360 o c) 60 o d) 45 o e) 120 o f) 15 o g) 135 o h) 270 o 2. Convet each of the following angles fom adians to degees. π a) adians b) 3π adians c) π adians d) π adians π e) 5π adians f) adians g) 7π adians h) π adians Convet each of the following angles fom degees to adians giving you answe to 2 decimal places. a) 17 o b ) 49 o c) 124 o d) 200 o 4. Convet each of the following angles fom adians to degees, giving you answe to 1 decimal place. a) 0.6 adians b) 2.1 adians c) 3.14 adians d) 1 adian 5. Finding an ac length when the angle is given in degees We know that if is measued in adians, then the length of an ac is given by s =. Suppose is measued in degees. We shall deive a new fomula fo the ac length. o s Figue 5. In this cicle the angle is measued in degees. Refeing to Figue 5, the atio of the ac length to the full cicumfeence will be the same as the atio of the angle subtended by the ac, to the angle in a full cicle; that is s 2π = 360 So, when is measued in degees we can use the following fomula fo ac length: s =2π 360 Notice how the ealie fomula, used when the angle is measued in adians, is much simple. 5 c mathcente June 11, 2004

6 6. The aea of a secto of a cicle A secto of a cicle with angle is shown shaded in Figue 6. Figue 6. The shaded aea is a secto of the cicle. The atio of the aea of the secto to the aea of the full cicle will be the same as the atio of the angle to the angle in a full cicle. The full cicle has aea π 2. Theefoe and so aea of secto aea of full cicle = 2π aea of secto = 2π π2 = Key Point aea of secto = when is measued in adians 7. Miscellaneous Examples Example Conside the cicle shown in Figue 7. Suppose we wish to calculate the angle. 25 Figue 7. Calculate the angle. c mathcente June 11,

7 We know the ac length and adius. We can use the fomula s =. Substituting the given values 25= and so = 25 =2.5 ads What is this angle in degees? We know π ads = 180 and so 1 ad = 180 π It follows that 2.5 ads = = π Example Refe to Figue 8. Suppose we have a cicle of adius cm and an ac of length 15cm. Suppose we want to find (a) the angle, (b) the aea of the secto OAB, (c) the aea of the mino segment (shaded). O B A 15 Figue 8. The shaded aea is called the mino segment. (a) Using s = we have 15 = and so = 15 =1.5c. (b) Using the fomula fo the aea of the secto, A = 1 2 2, we find aea = = 1 2 (2 )(1.5) = 75 cm 2 (c) We aleady know that the aea of the secto OAB is 75cm 2. If we can wok out the aea of the tiangle AOB we can then detemine the aea of the mino segment. (Recall the fomulae fo the aea of tiangle, A = 1 ab sin C.) 2 aea of tiangle = sin = sin 1.5 = cm 2 7 c mathcente June 11, 2004

8 Theefoe the aea of the mino segment is = cm 2 (to 2 dp.) Example Suppose we have an angle of 120. What is this angle in adians? We know that π ads = 180 and so then This can be witten as 2π 3 π ads = = π 120 ads 180 adians (= adians). Execise 3 A secto of a cicle is an aea bounded by two adii and an ac. A secto has an angle at the cente of the cicle. All the questions below elate to a cicle with adius 5cm. 1. Detemine the length of the ac (coect to 2 decimal places) when the angle at the cente is a) 1.2 adians b) π adians c) 45o 2 2. Calculate the aea (coect to 2 decimal places) of each of the thee sectos in Question A secto of this cicle has aea 50 cm 2. What is the angle (in adians) at the cente of this secto? Answes Execise 1 a) 2 b) π c) 0.5 d) 2π Execise 2 1. a) 1 b) 2 c) 1 d) 1 e) 2 f) 1 g) 3 h) a) 90 o b) 135 o c) 180 o d) 30 o e) 900 o f) 144 o g) 315 o h) 18 o 3. a) 0.30 adians b) 0.86 adians c) 2.16 adians d) 3.49 adians 4. a) 15.3 o b) o c) o d) 57.3 o Execise 3 1. a) 6 cm b) 7.85 cm c) 3.93 cm 2. a) 15 cm 2 b) cm 2 c) 9.82 cm adians c mathcente June 11,

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