The Other Trigonometric Functions

The Other Trigonometric Functions tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩn...

Another .: môt .nào đó số ít ,dùng khi nói đến một đối tượng nào đó không xác định This book is boring .Give me another quyển sách này chán quá đưa tôi quyển khác xem => quyển nào cũng được ,không xác định Others : những khác Số nhiều ,dùng khi nói đến những đối tượng nào đó không xác định These books are boring .Give me others : những quyển sách này chán quá ,đưa tôi những quyển khác xem => tương tự câu trên nhưng số nhiều the other : .còn lại Xác định ,số ít I have two brothers .One is a dotor ; the other is a teacher Tôi có 2 ngừoi anh .Một ngừoi là bác sĩ ngừoi còn lại là giáo viên. the others : những .còn lại Xác định ,số nhiều I have three brothers .One is a dotor ; the others are teachers Tôi có 3 ngừoi anh .Một ngừoi là bác sĩ những ngừoi còn lại là giáo viên. The others = The other + N số nhiều There are 5 books on the table .I don't like this book .I like the others = ( I like the other books ) http://www.tienganh.com.vn/showthread.php?p=144147#post144147 The Other Trigonometric Functions The Other Trigonometric Functions By: OpenStaxCollege A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 12 or less, regardless of its length A tangent represents a ratio, so this means that for every inch of rise, the ramp must have 12 inches of run Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions We have already defined the sine and cosine functions of an angle Though sine and cosine are the trigonometric functions most often used, there are four others Together they make up the set of six trigonometric functions In this section, we will investigate the remaining functions Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent To define the remaining functions, we will once again draw a unit circle with a point (x, y) corresponding to an angle of t, as shown in [link] As with the sine and cosine, we can use the (x, y) coordinates to find the other functions The first function we will define is the tangent The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle In [link], y the tangent of angle t is equal to x , x≠0 Because the y-value is equal to the sine of t, 1/28 The Other Trigonometric Functions and the x-value is equal to the cosine of t, the tangent of angle t can also be defined as sin t cos t , cos t ≠ 0.The tangent function is abbreviated as tan The remaining three functions can all be expressed as reciprocals of functions we have already defined • The secant function is the reciprocal of the cosine function In [link], the secant 1 of angle t is equal to cos t = x , x ≠ The secant function is abbreviated as sec • The cotangent function is the reciprocal of the tangent function In [link], the cos t x cotangent of angle t is equal to sin t = y , y ≠ The cotangent function is abbreviated as cot • The cosecant function is the reciprocal of the sine function In [link], the 1 cosecant of angle t is equal to sin t = y , y ≠ The cosecant function is abbreviated as csc A General Note Tangent, Secant, Cosecant, and Cotangent Functions If t is a real number and (x, y) is a point where the terminal side of an angle of t radians intercepts the unit circle, then y tan t = , x ≠ x sec t = , x ≠ x csc t = , y ≠ y x cot t = , y ≠ y Finding Trigonometric Functions from a Point on the Unit Circle ( ) The point − √2 , is on the unit sin t, cos t, tan t, sec t, csc t, and cot t circle, as shown in [link] Find 2/28 The Other Trigonometric Functions Because we know the (x, y) coordinates of the point on the unit circle indicated by angle t, we can use those coordinates to find the six functions: sin t = y = cos t = x = − √ ( ) y tan t = = = − = − = −√ √ x √3 √3 − sec t = = x csc t = 1 = =2 y − √2 = − 2 = − √ √3 − √2 x cot t = = = − √ = − √3 y () Try It ( 2 ) The point √2 , − √2 is on the unit sin t, cos t, tan t, sec t, csc t, and cot t circle, as shown in [link] Find 3/28 The Other Trigonometric Functions sin t = − √2 , cos t = √2 , tan t = − 1, sec t = √2, csc t = − √2, cot t = − Finding the Trigonometric Functions of an Angle π Find sin t, cos t, tan t, sec t, csc t, and cot t when t = We have previously used the properties of equilateral triangles to demonstrate that π π sin = and cos = √2 We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values π π sin tan = cos π = √3 = =√ √3 = 2 = √ √3 π sec = cos π = √3 4/28 The Other Trigonometric Functions π 1 csc = = =2 sin π π π cos cot = sin π √3 = 2 = √3 Try It π Find sin t, cos t, tan t, sec t, csc t, and cot t when t = π sin = π cos = √3 2 π tan = √3 π sec = π 2√3 π √3 csc = cot = Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x equal to the cosine and y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent The results are shown in [link] Angle π 6, Cosine √3 √2 2 Sine √2 √3 Tangent √3 √3 Undefined Secant 2√3 √2 Undefined Cosecant Undefined √2 2√3 1 √3 Cotangent Undefined √3 or 30° π 4, or 45° π 3, or 60° π 2, or 90° 5/28 The Other Trigonometric Functions Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant [link] shows which functions are ... DECLARATION I certify that all the material in this study which is not my own work has been identified and acknowledged, and that no material is included for which a degree has already been conferred upon me. Signature Ta Thi Minh Nguyet i ACKNOWLEDGEMENTS I would like to acknowledge a number of individuals without whose help this study would have been impossible. First of all, I would like to express my indebtedness to my supervisor, Ms. Phan Hoang Yen, for her very invaluable advice as well as great help in the completion of this study. Secondly, I am also very grateful to my friends, the teachers and the students of linguistics at USSH - VNU for their precious suggestions, comments and cooperation. Last but not least, my sincerest thanks go to my family, whose great support and encouragement have helped me a lot in the production of this study. To all these people, many thanks again. I can only hope that they will look upon the results of their influence and endeavors with pleasure. ii ABSTRACT It is undeniable that English plays an important part in one’s professional career. However, in Department of Linguistics and Vietnamese Studies at University of Social Sciences and Humanities, Vietnam National University, although English has long been introduced as a compulsory subject, the teaching and learning of ESP in general and reading skills in particular are still far from being satisfactory. To be more exact, the Communicative Approach is not properly applied and reading classes are often used to teach language rather than reading comprehension. With the desire to overcome the exiting problems and improve all the four language skills in reading lessons in ESP for linguistics, this study is carried out. It consists of three parts. Part A is about the rationale, objectives, scope and methods of the study. Part B deals with the theoretical background knowledge. It is divided into two chapters. − Chapter I presents an overall view of reading and reading in ESP teaching and learning; − Chapter II introduces the advantages of skill-integration in teaching and learning reading and relationship between this skill and the other language skills. Part C is the main part of the study. It consists of two chapters. − Chapter III aims at identifying and analyzing both strong points and weak points of the teaching and learning of reading ESP in Department of Linguistics and Vietnamese Studies at USSH – VNU based on the two survey questionnaires and the observation. The findings, comments and conclusions are also given; − Chapter IV suggests some techniques for each stage of a reading lesson. They are carried out in the light of Communicative Language Teaching with a hope that there will be some improvement in teaching and learning reading skills in integration with the development of the other language skills. Part D summarizes what have been presented and discussed in the study and gives some suggestions    –: the other post-Kantian: Jacob Friedrich Fries and non-Romantic Sentimentalism Although Romanticism dominated the development of immediate post-Kantian thought (after Reinhold), there were other, equally im- portant interpretations afoot of where to take Kant. By the turn of the century (), Jacobi’s influence, always large in this period, had al- ready led to another, very different, appropriation of Kant in the per- son of Jacob Friedrich Fries (–). About the same age as the other post-Kantians at Jena (Schelling, Hegel, Schleiermacher, Novalis, and H¨olderlin), Fries only managed to formulate his own views about a decade later than those working in the aftermath of the initial tumult surrounding Fichte and the early Romantics. Like many of them (for ex- ample, Niethammer, H¨olderlin, Schelling, Hegel, and Schleiermacher), he too had first studied theology before moving to philosophy. Having been raised and educated in a famous Pietist community of the Herrnhut (Moravian) Brethren, he was sent to a Pietist boarding school in Niesky for his adolescent years. In , he went to Leipzig to study philosophy, where he apparently came under the influence of Jacobi’s work; in , he studied for a year in Jena, leaving for while to be a private tutor, only to return to Jena at the end of  (around the same time Hegel arrived in Jena). After , he and Jacobi became friends, and Jacobi remained an admirer of Fries’s work. Fries’s own career was rather checkered, and he and Hegel developed a distaste for each other at Jena that spanned the lifetimes of both men, leading both to denounce each other in private and public in a variety of ways for their entire lives. Fries nonetheless established his views as one of the major options in the post-Kantian debate, and, in many ways, Fries, Schelling, and Hegel contended for preeminence in the German philosophical scene during the lives of all three men. Like many other men of his generation, Fries found his academic job prospects rather paltry (although he was far more successful at first than Hegel), and he bitterly resented others attaining any of the few positions available   Part II The revolution continued: post-Kantians ( just as Hegel, and others, bitterly resented Fries’s own acquisition of any of the few positions that were available). Fries was quite industrious and, starting around , published vol- ume after volume laying out his own system of post-Kantian thought. His own entry into the scene came in  with the publication of Reinhold, Fichte, Schelling, which sharply criticized all three thinkers and established his own views as being markedly different from all the other versions of “idealism” being touted around Jena at the time. (In some ways, that book can be seen as his own riposte to Hegel’s first book in , The Difference Between Fichte’s and Schelling’s Systems of Philosophy.) In the same year, he published his Philosophical Doctrine of Right and Critique of All Positive Legislation,in his first presentation of his complete system as Knowledge, Faith, and Portent, and in  his multi-volume NewCritique of Reason, which he then revised and republished later in – as the Newor Anthropological Critique of Reason. His position, however, was already set out in its basic formby  with the publication of Reinhold, Fichte, Schelling, and, in his other writings, he tended to repeat himself quite a bit.  Fries TABLE Natural Trigonometric Functions 1′ sec Diff 1′ cot 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 30 Diff 1′ ∞ 3437.75 1718.87 1145.92 859.436 687.549 572.957 491.106 429.718 381.971 343.774 312.521 286.478 264.441 245.552 229.182 214.858 202.219 190.984 180.932 171.885 163.700 156.259 149.465 143.237 137.507 132.219 127.321 122.774 118.540 114.589 110.892 107.426 104.171 101.107 98.2179 95.4895 92.9085 90.4633 88.1436 85.9398 83.8435 81.8470 79.9434 78.1263 76.3900 74.7292 73.1390 71.6151 70.1533 68.7501 67.4019 66.1055 64.8580 63.6567 62.4992 61.3829 60.3058 59.2659 58.2612 57.2900 tan — — 1718.88 572.958 286.479 171.887 114.592 81.851 61.388 47.747 38.197 31.252 26.044 22.037 18.889 16.370 14.324 12.639 11.235 10.052 9.047 8.185 7.441 6.794 6.228 5.730 5.289 4.897 4.547 4.234 3.952 3.697 3.466 3.256 3.064 2.8890 2.7285 2.5810 2.4452 2.3198 2.2038 2.0963 1.9965 1.9036 1.8171 1.7363 1.6608 1.5902 1.5239 1.4617 1.4033 1.3482 1.2964 1.2475 1.2013 1.1576 1.1162 1.0771 1.0399 1.0047 0.9712 Diff 1′ sec 1.00000 00000 00000 00000 00000 1.00000 00000 00000 00000 00000 1.00000 00001 00001 00001 00001 1.00001 00001 00001 00001 00002 1.00002 00002 00002 00002 00002 1.00003 00003 00003 00003 00004 1.00004 00004 00004 00005 00005 1.00005 00005 00006 00006 00006 1.00007 00007 00007 00008 00008 1.00009 00009 00009 00010 00010 1.00011 00011 00011 00012 00012 1.00013 00013 00014 00014 00015 1.00015 csc Diff 1′ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 Diff 1′ cos 1.00000 00000 00000 00000 00000 1.00000 00000 00000 00000 00000 1.00000 99999 99999 99999 99999 0.99999 99999 99999 99999 99998 0.99998 99998 99998 99998 99998 0.99997 99997 99997 99997 99996 0.99996 99996 99996 99995 99995 0.99995 99995 99994 99994 99994 0.99993 99993 99993 99992 99992 0.99991 99991 99991 99990 99990 0.99989 99989 99989 99988 99988 0.99987 99987 99986 99986 99985 0.99985 sin Diff 1′ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 ′ 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 Diff 1′ 89° 1°➙ ′ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 sin Diff 1′ csc Diff 1′ 0.01745 01774 01803 01832 01862 0.01891 01920 01949 01978 02007 0.02036 02065 02094 02123 02152 0.02181 02211 02240 02269 02298 0.02327 02356 02385 02414 02443 0.02472 02501 02530 02560 02589 0.02618 02647 02676 02705 02734 0.02763 02792 02821 02850 02879 0.02908 02938 02967 02996 03025 0.03054 03083 03112 03141 03170 0.03199 03228 03257 03286 03316 0.03345 03374 03403 03432 03461 0.03490 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 30 29 29 29 29 29 29 57.2987 56.3595 55.4505 54.5705 53.7179 52.8916 52.0903 51.3129 50.5584 49.8258 49.1141 48.4224 47.7500 47.0960 46.4596 45.8403 45.2372 44.6498 44.0775 43.5196 42.9757 42.4452 41.9277 41.4227 40.9296 40.4482 39.9780 5185 39.0696 38.6307 38.2016 37.7818 37.3713 36.9695 5763 36.1914 35.8145 4454 35.0838 34.7295 34.3823 34.0420 33.7083 3812 33.0603 32.7455 4367 32.1337 31.8362 5442 31.2576 30.9761 6996 4280 30.1612 29.8990 6414 3881 29.1392 28.8944 28.6537 9392 9089 8801 8526 8263 8013 7774 7545 7326 7117 6917 6724 6540 6363 6194 6031 5874 5723 5578 5439 5305 5175 5051 4930 4814 4702 4594 4490 4389 4291 4197 4106 4017 3932 3849 3769 3691 3616 3543 3472 3403 3336 3272 3209 3148 3088 3031 2974 2920 2867 2815 2765 2716 2668 2622 2577 2532 2490 2448 2407 ➙ Diff 0.00000 00029 00058 00087 00116 0.00145 00175 00204 00233 00262 0.00291 00320 00349 00378 00407 0.00436 00465 00495 00524 00553 0.00582 00611 00640 00669 00698 0.00727 00756 00785 00815 00844 0.00873 00902 00931 00960 00989 0.01018 01047 01076 01105 01135 TABLE cot Diff 1′ 10.00000 00000 00000 00000 00000 10.00000 00000 00000 00000 00000 10.00000 00000 00000 00000 00000 10.00000 00000 00001 00001 00001 10.00001 00001 00001 00001 00001 10.00001 00001 00001 00001 00002 10.00002 00002 00002 00002 00002 10.00002 00002 00003 00003 00003 10.00003 00003 00003 00003 00004 10.00004 00004 00004 00004 00004 10.00005 00005 00005 00005 00005 10.00006 00006 00006 00006 00006 10.00007 tan csc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Diff 1′ 10.00000 00000 00000 00000 00000 10.00000 00000 00000 00000 00000 10.00000 00000 00000 00000 00000 10.00000 00000 99999 99999 99999 9.99999 99999 99999 99999 99999 9.99999 99999 99999 99999 99998 9.99998 99998 99998 99998 99998 9.99998 99998 99997 99997 99997 9.99997 99997 99997 99997 99996 9.99996 99996 99996 99996 99996 9.99995 99995 99995 99995 99995 9.99994 99994 99994 99994 99994 9.99993 179° sin 1° ➙ ′ 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 ′ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 89° sin 8.24186 24903 25609 26304 26988 8.27661 28324 28977 29621 30255 8.30879 31495 32103 32702 33292 8.33875 34450 35018 35578 36131 8.36678 37217 37750 38276 38796 8.39310 39818 40320 40816 41307 8.41792 42272 42746 43216 43680 8.44139 44594 45044 45489 45930 8.46366 46799 47226 47650 48069 8.48485 48896 49304 49708 50108 8.50504 50897 51287 51673 52055 8.52434 52810 53183 53552 53919 8.54282 91°➙ cos Diff 1′ csc tan Diff 1′ cot sec 717 706 695 684 673 663 653 644 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 514 508 502 496 491 485 480 474 470 464 459 455 450 445 441 436 433 427 424 419 416 411 408 404 400 396 393 390 386 382 379 376 373 369 367 363 11.75814 75097 74391 73696 73012 11.72339 71676 71023 70379 69745 11.69121 68505 67897 67298 66708 11.66125 65550 64982 64422 63869 11.63322 62783 62250 61724 61204 11.60690 60182 59680 59184 58693 11.58208 57728 57254 56784 56320 11.55861 55406 54956 54511 54070 11.53634 53201 52774 52350 51931 11.51515 51104 50696 50292 49892 11.49496 49103 48713 48327 47945 11.47566 47190 46817 46448 46081 11.45718 8.24192 24910 25616 26312 26996 8.27669 28332 28986 29629 30263 8.30888 31505 32112 32711 33302 8.33886 34461 35029 35590 36143 8.36689 37229 37762 38289 38809 8.39323 39832 40334 40830 41321 8.41807 42287 42762 43232 43696 8.44156 44611 45061 45507 45948 8.46385 46817 47245 47669 48089 8.48505 48917 49325 49729 50130 8.50527 50920 51310 51696 52079 8.52459 52835 53208 53578 53945 8.54308 718 706 696 684 673 663 654 643 634 625 617 607 599 591 584 575 568 561 553 546 540 533 527 520 514 509 502 496 491 486 480 475 470 464 460 455 450 446 441 437 432 428 424 420 416 412 408 404 401 397 393 390 386 383 380 376 373 370 367 363 11.75808 75090 74384 73688 73004 11.72331 71668 71014 70371 69737 11.69112 68495 67888 67289 66698 11.66114 65539 64971 64410 63857 11.63311 62771 62238 61711 61191 11.60677 60168 59666 59170 58679 11.58193 57713 57238 56768 56304 11.55844 55389 54939 54493 54052 11.53615 53183 52755 52331 51911 11.51495 51083 50675 50271 49870 11.49473 49080 48690 48304 47921 11.47541 47165 46792 46422 46055 11.45692 10.00007 00007 00007 00007 00008 10.00008 00008 00008 00008 00009 10.00009 00009 00010 00010 00010 10.00010 00011 00011 00011 00011 10.00012 00012 00012 00013 00013 10.00013 00014 00014 00014 00015 10.00015 00015 00016 00016 00016 10.00017 00017 00017 00018 00018 10.00018 00019 00019 00019 00020 10.00020 00021 00021 00021 00022 10.00022 00023 00023 00023 00024 10.00024 00025 00025 00026 00026 10.00026 Diff 1′ sec cot Diff 1′ tan csc Diff 1′ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 Diff 1′ cos 178° ′ 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9.99993 ... what is the cot(−t) ? Graphical For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions 24/28 The Other Trigonometric Functions. .. and cotangent functions is π Finding the Values of Trigonometric Functions Find the values of the six trigonometric functions of angle t based on [link] 16/28 The Other Trigonometric Functions sin... to Relate Trigonometric Functions 13/28 The Other Trigonometric Functions 12 If cos(t) = 13 and t is in quadrant IV, as shown in [link], find the values of the other five trigonometric functions