2. Trigonometric Functions

Transcription

1 . Tigonometic Functions. Radians Definition A adian is a measue of angle size. It is defined by the diagam at the ight, that is, adian is the angle which subtends an ac of length in a cicle of adius. Example.How big is a adian in degees? Solution. Hee s a nice agument that it s just less than 60. If we wee to spead the two adii just enough to staighten out that piece of ac of length, the angle between them would then be a bit geate than adian. But then we would have a tiangle with all sides of length, hence all angles would be 60. Thus the oiginal adian must be somewhat less than 60. But thee is a good agument that establishes an exact answe. If adian subtends an ac of, then adians (in the same cicle) should subtend an ac of, and adians should subtend an ac of, etc. Quite geneally, adians should subtend an ac of length. This "popotionality" obsevation elies on the adial symmety of the cicle. Thus, in a cicle of adius, the angle in adians and the ac length ae the same. Now thee is one ac whose length we know, and that's the whole cicumfeence of the cicle, and it has length C = π = π = π. It follows that a whole evolution (60 ) must be π adians. Thus: π adians = 60 degees 60 adian = degees π = 7. (appoximately) A numbe of students have touble with the notion that the above diagam could seve as a definition of a adian. To solve Example below, they want to know exactly how big a adian is (say in degees?) No. The diagam is the definition. And an elegant definition it is too. It convinces us that the adian must be a natual measue of angle. π π By taking a few special factions of 60, we get the following useful coespondences. 60 degees = π adians degees = π adians 90 degees = π/ adians 4 degees = π/4 adians 60 degees = π/ adians 0 degees = π/6 adians adians 8/6/007

2 Ac length Example. What size of ac is subtended by an angle of adians in a cicle of adius? That is, what is x? This is a good poblem in that thee ae two changes between the fist pictue (the definition) and this one--the size of the angle and the adius of the cicle, and each of these has an effect on the ac length. In fact what one ought to do is take the poblem apat and analyze these effects one at a time. And we can take them in eithe ode too. In fact the elegant diagam below nicely summaizes both outes. If we genealize this agument, we get a fomula fo any angle in any cicle. The ac subtended by an angle of adians in a cicle of adius is a =. 6 x The two aows going up to the ight use the fact that if you blow up a pictue by a facto of, all lengths ae multiplied by. The two aows going down to the ight use the fact that in a cicle of fixed size, ac length is popotional to angle double the angle and you double the ac. The agument that concludes that x=6 can un eithe along the top oute o the bottom. Ac-length fomula: a = Hee, the angle has to be in adians. The agument is given at the ight. The upaows use the fact that if you blow up a pictue all lengths change by the same facto, and the down-aows use the fact that in a cicle of fixed size, ac length is popotional to angle. adians 8/6/007

3 Aea of secto Example. What is the aea A of the secto fomed by an angle of adians in a cicle of adius? Solution. Again we use a popotionality agument. The adial symmety of the cicle tells us that in a fixed cicle, the aea of the secto will be popotional to the cental angle. A Again, thee is one aea which we know, and that's the whole cicle, and it has aea π. This coesponds to an entie evolution which has angle π adians. Thus: π adians gives aea π π adian gives aea = π Aea of secto A = Again the angle has to be in adians. adians gives aea A = This is not a fomula to be memoized. The agument at the ight is impotant enough and simple enough that it should always be epeated. Example 4. Find the aea A of the secto fomed by an angle of in a cicle of adius. Solution. We have a geneal fomula sitting up thee which we could use if we conveted ou angle to adian measue. Altenatively we could develop a fomula valid fo degee measue. But fomulas can be dead weight. The best oute is to simply use the popotionality idea diectly. 60 gives aea π = π gives aea π 60 gives aea A = π = A A secto is a egion of a cicle bounded by two adii and the ac between them. A segment is a egion of a cicle bounded by a chod and the ac it subtends. Fo example, the egion B in Example below is a segment. adians 8/6/007

4 Example. Find the aea B of the segment shown in the diagam at the ight. Solution. In Example 4 we have calculated the aea A of the whole secto. We can get the aea of the segment by subtacting fom A the aea T of the tiangle. B Now the tiangle is isosceles, so it's nice to otate it and make the chod the base. To find its aea we eect an altitude which bisects the angle. The semibase is then sin40 and the altitude is cos40. The aea of each sub-tiangle will be half the poduct of these two and so the aea of the whole tiangle will be the whole poduct: T = (sin40)(cos40) =. The segment aea is then: B = A T = π sin40cos40 = 7.4. =.4 60 T cos40 sin40 Poblems. Find exact expessions fo the following. Use adian measue fo the tig functions. (a) sin(π/) (b) sin(π/) (c) sin(4π/) (d) sin(π) (e) cos(π) (f) sin(π/4) (g) cos(π/4). What angle has the popety that its size in degees exceeds its size in adians by 00?. What length of ac is subtended by an angle of 0. adians in a cicle of adius? 4. What length of ac is subtended by an angle of degees in a cicle of adius?. What length of chod is subtended by an angle of degees in a cicle of adius? 6. What angle will subtend an ac of length in a cicle of adius 4? 7. What is the aea A of the secto fomed by an angle of. adians in a cicle of adius 6? 8. Find the aea A of the secto fomed by an angle of 0 in a cicle of adius 0. adians 8/6/007 4

5 9. In a cicle of adius, a chod has length 6. What is the aea of the segment defined by the chod? 0. Two stakes ae put in the gound 8 m apat and two equally feocious goats ae tetheed, one on each stake, on opes of length m. How much gass does each goat get, assuming they each get an equal shae of the common gound? Assume the gass has density kg/m..(a) Find the length of the ac cut off the cicle x + y = by the line y = 0 x. (b) Find the aea of the segment cut off the cicle x + y = by the line y = 0 x.. If I daw two intesecting cicles of adius, I fom thee closed egions. If they all have the same aea, how fa apat ae the centes of the two cicles?. Define the functions f(x) = sinx whee x is measued in adians g(x) = sinx whee x is measued in degees. On the same set of axes, daw the gaphs of both f(x) and g(x) fo 0 x 0. Indicate on you gaph the smallest positive solution to the equation f(x) = g(x) and estimate the numeical value of this x. Of couse this is an amusing value of x. It has the popety that if you ask me fo its sine, I don t have to ask you whethe to put my calculato in adian o degee mode. The answe s the same! This eminds me of the Centigade and Fahenheit tempeatue scales. If you tell me it s 40 below I don t have to ask you whethe you mean C o F. adians 8/6/007