Lecture Notes Trigonometric Limits page 1

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1 Lecture Notes Trigoometric Limits age Theorem : si! Proof: This theorem ad the et oe are ecessary for di eretiatig si ad cos. Recall a theorem: Let r be the radius of a circle. If is measured i radias, the the area of a sector with a cetral agle of is A sector r. (Notatio: AB will deote the legth of lie segmet AB.) Let be a very small ositive agle, measured i radias, draw ito a uit circle as show o the icture below. Let B be the oit where the uit circle itersects the ray determied by. We the draw a taget lie to the circle at oit B. Let A be the oit where the taget lie itersects the ais. We also draw a vertical lie through B: Let D be the oit where this vertical lie itersects the ais. Fially, let us deote by E the oit with coordiates (; ). The roof will be based o the followig fact: because they iclude each other, the followig three areas ca be easily comared: Area of triagle CDB Area of sector CEB Area of triagle ABC Area of triagle CDB: the horizotal side, CD cos ad the vertical side, DB si. Sice this is a right triagle, the area is: A CDB si cos Area of sector CEB: A sector Area of triagle ABC: there is a right agle at oit B because the taget lie draw to a circle is eredicular to the radius draw to the oit of tagecy. So the area is A ABC AB BC. Clearly BC. To comute AB, i triagle ABC, ta AB ad so AB ta. Area of triagle ABC: ta () (ta ) or si. So ow cos Area of triagle CDB Area of sector CEB Area of triagle ABC traslates to Let us divide all three sides by si reverse the iequality sigs. si cos si cos. Because is small ad ositive, si cos si cos is ositive ad so we do ot eed to c Hidegkuti, Last revised: Jue 7, 5

2 Lecture Notes Trigoometric Limits age Suose ow that aroaches zero. The both cos ad cos aroach. By the sadwich ricile, si, the quatity locked i betwee those two must also aroach : If si aroaches ; so does its recirocal, si. cos si cos # # So far, we have rove the statemet for ositive values of, that is, for egative values of. si! +. A similar argumet works Theorem : cos! Proof: cos! cos cos cos + cos!! cos +! (cos + )! si! (cos + ) si! si cos + si! si! cos + cos (cos + ) Samle Problems si. Recall that. Use this fact to comute each of the followig its.! a)! si 5 b)! cos c) cos! d) ta! e) si 5! si 6 f)! g)! ta cos. a) Fid the erimeter of a 5 sided reguar olygo writte ito a circle with radius m. b) Fid the erimeter of a sided regular olygo writte ito a circle with radius R. Use radias to measure agles. c) Fid the it of the erimeter of a sided regular olygo writte ito a circle with radius R as aroaches i ity. Use radias to measure agles. 3. a) Fid the area of a 5 sided regular olygo writte ito a circle with radius m. b) Fid the area of a sided regular olygo writte ito a circle with radius R. Use radias to measure agles. c) Fid the it of the area of a sided regular olygo writte ito a circle with radius R as aroaches i ity. Use radias to measure agles. c Hidegkuti, Last revised: Jue 7, 5

3 Lecture Notes Trigoometric Limits age 3 Practice Problems Comute each of the followig its..! si 5.! si cos 3.! cos 3 si 3.! si + si 5.! ta 5 6.! si si 3 7.! si 3 si 8.! 3 si 9.! ta.! ta 6 3.! si ta.! si ta 3 3.! si cos.! si 5.! ta ta 6.! 6 si + si 3 7.! cos 8.! si 9.! si cos Samle Problems - Aswers. a) 5 b) c) d) e) 5 f) g) udeed 6. a) 3 si m 6: 3735 m b) R si 5 c) R 3. a) 75 si m 35: 55 m b) 5 R si c) R Aswers - Practice Problems.) 5.) 3.) 3.) 6 5.) 5 6.) 3 7.) 3 8.) 3 9.).).).) 6 3.).) 5.) 6.) ) 8.) 9.) c Hidegkuti, Last revised: Jue 7, 5

4 Lecture Notes Trigoometric Limits age Samle Problems - Solutios si. Recall that. Use this fact to comute each of the followig its.! si 5 a)! si 5 si 5 si 5 5!!! 5 si 5! si 5! 5 Let y 5: As aroaches zero, so does y. b)! cos cos! c)! cos cos + cos! So the it becomes si 5 5! 5 5 si y y! y + cos! si! si! + cos cos cos!! + cos si!! + cos 5 5 cos ( + cos )! + cos cos! ( + cos )! si!! si si ( + cos )! si + cos si si ( + cos )! + cos + cos d)! ta ta!! si cos si si! cos! si cos!! cos e)! si 5 si 6 Solutio: We will brig this it to a form where si aears. si 5 si 5! si 6! si 6 si 5 si 5! si 6!! si 6 si 5 5! 5! si 6 6 si 5 6 6! 5 5! si 6 6 si 5 5! 5 6 6! si c Hidegkuti, Last revised: Jue 7, 5

5 Lecture Notes Trigoometric Limits age 5 f)! ta This is clearly a tye of a idetermiate. To simlify the deomiator, we will itroduce a ew variable. Let. As aroaches, will aroach zero. Also, solvig for we get + : So our it becomes ta ta +!! Now the deomiator is simle, but the umerator became more comle. We will ead ta + usig the sum formula for taget. ta +! ta + ta ta ta!! ta + ( ta )! ta! ta ta! ta + ta!! ta + + ta ta! ta ta ta + ta ta ta! ta ta g)! cos This is also a tye of a idetermiate. Solutio : We will start by multilyig both umerator ad deomiator by + cos.! cos!! cos cos + cos! + cos + cos! si + cos! cos + cos + cos jsi j + cos! ( cos ) ( + cos ) + cos si If the eressio was simly si, the we would be i a good shae, sice. But we + cos! ca ot igore that we have the absolute value of si. We get rid of the absolute value sig by cosiderig the sig of si : Case. If >, the also si > ad so jsi j si! + cos jsi j! + + cos si! + + cos si! +! + + cos Case. If <, the also si < ad so jsi j si! cos! jsi j + cos! si + cos! si! + cos Sice the left-had side it ad the right-had side it are di eret, the two-sided it is ude ed. c Hidegkuti, Last revised: Jue 7, 5

6 Lecture Notes Trigoometric Limits age 6 Solutio : Recall that si cos! cos ad so cos si.! r si si! We will eed to be a little bit careful because of the absolute value. If is ositive (recall it is also very close to zero) the so is si. its. Let us itroduce the ew variable y :! + si! + If is egative, so is si. We will searately evaluate the left-side ad right-side si si y y! + y! + si The other side goes similarly: si si!! +! + si! + si! + si si y y! + y si! +! + si! + si! + si Sice the right-had side it ad the left-had side it are di eret, the two-sided it is ude ed.. a) Fid the erimeter of a 5 sided reguar olygo writte ito a circle with radius m. The agle at the ceter of the circle is 36 5 If we draw the altitude belogig to side, we create a right triagle with a agle of. From this right triagle, si r r. So r si. We covert the agle to radias ad substitute m for r: The erimeter is the sum of all 5 sides: P 5 5 (r si ) 3 ( m) si 3 si m 6: m 5 5 c Hidegkuti, Last revised: Jue 7, 5

7 Lecture Notes Trigoometric Limits age 7 b) Fid the erimeter of a sided regular olygo writte ito a circle with radius R. Use radias to measure agles. Solutio: We will erform the same stes as i the revious roblem, oly i the abstract. R si R si. Ad so the erimeter of the olygo is P R si R si c) Fid the it of the erimeter of a sided regular olygo writte ito a circle with radius R as aroaches i ity. Use radias to measure agles. R si B C B A si!!! C A R A! De e. si As! ; clearly! : Thus si A R! si R R 3. a) Fid the area of a 5 sided regular olygo writte ito a circle with radius si Solutio: Oe ca comute the altitude of the right triagle usig right triagle trigoometry, ad comute the area that way. However, we will use a moe e ciet techique. Recall that the area of a triagle ca be comuted as A ab si where iis the agle betwee sides a ad b: The we ca immediately comute the area of the isosceles triagle: R si. So the area of the olygo is A 5 R si 5 ( m) si 75 si m 35: 55 m 5 5 b) Solutio: We will erform the same stes as i the revious roblem, oly i the abstract. A R si R si c) Fid the it of the area of a sided regular olygo writte ito a circle with radius R as aroaches i ity. Use radias to measure agles.! R si B C B si C A R B A!!! si De e. As! ; clearly! : Thus R! si C A R! si R R For more documets like this, visit our age at htt:// ad click o Lecture Notes. questios or commets to mhidegkuti@ccc.edu. c Hidegkuti, Last revised: Jue 7, 5

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