,.)"Ji . ',""" ., " TRIGONOMETRIC FUNCTIONS / l II s, A. A. I1aHlJRmKlIH, E. T. llIaBryJIR)J,3e TPl1rOHOMETPI1QECRI1E <DYHRUl1lJ B 3A,D;AQAX ~I3AaTeJlhCTHO «Hayrca: MOCRBa ~ I 1 , 1 I -, I A. Panchishkin E.Shavgulidze ' TRIGONOMETRIC FUNCTIONS (Problem-50lving Approach) Mir Publishers Moscow' Translated from Russian by Leonid Levant First published 1988 Revised from the 1986 Russian edition Ha anaAuiic1:0M nsune Printed in the Union of Soviet Socialist Republics I; t /1 I r I ISBN 5-03-000222-7 © 1I3p;aTeJIhcTBo «HayKa., I'aaanaa penasnaa .pHaHKO-MaTeMaTHQeCKOii nareparypu, 1986 © English translation, Mir Publishers, 1988 FroD1 the J\uthors By tradition, trigonometry is an important component of mathematics courses at high school, and trigonometry questions are always set at oral and written examina- tions to those entering universities, engineering colleges, and teacher-training institutes. The aim of this study aid is to help the student to mas- ter the basic techniques of solving difficult problems in trigonometry using appropriate definitions and theorems from the school course of mathematics. To present the material in a smooth way, we have enriched the text with some theoretical material from the textbook Algebra and Fundamentals of Analysis edited by Academician A. N. Kolmogorov and an experimental textbook of the same title by Professors N.Ya. Vilenkin, A.G. Mordko- vich, and V.K. Smyshlyaev, focussing our attention on the application of theory to solution of problems. That is why our book contains many worked competition problems and also some problems to be solved independ- ently (they are given at the end of each chapter, the answers being at the end of the book). Some of the general material is taken from Elementary Mathematics by Professors G.V. Dorofeev, M.K. Potapov, and N.Kh. Rozov (Mir Publishers, Moscow, 1982), which is one of the best study aids on mathematics for pre- college students. We should like to note here that geometrical problems which can be solved trigonometrically and problems involving integrals with trigonometric functions are not considered. At present, there are several problem hooks on mathe- matics (trigonometry included) for those preparing to pass their entrance examinations (for instance, Problems 6 From the Authors at Entrance Examinations in Mathematics by Yu.V. Nes- terenko, S.N. Olekhnik, and M.K. Potapov (Moscow, Nauka, 1983); A Collection of Competition Problems in Mathematics with Hints and Solutions edited by A.I. Pri- Iepko (Moscow, Nauka, 1986); A Collection of Problems in Mathematics for Pre-college Students edited by A. I. Pri- lepko (Moscow, Vysshaya Shkola, 1983); A Collection of Competition Problems in Mathematics for Those Entering Engineering Institutes edited by M.1. Skanavi (Moscow, Vysshaya Shkola, 1980). Some problems have been bor- rowed from these for our study aid and we are grateful to their authors for the permission to use them. The beginning of a solution to a worked example is marked by the symbol and its end by the symbol ~. The symbol ~ indicates the end of the proof of a state- ment. Our book is intended for high-school and pre-college students. We also hope that it will be helpful for the school children studying at the "smaller" mechanico- mathematical faculty of Moscow State University. From the Authors Contents 5 Chapter 1. Definitions and Basic Properties of Trigono- metric Functions 9 1.1. Radian Measure of an Arc. Trigonometric Circle 9 1.2. Definitions of the Basic Trigonometric Func- tions 18 1.3. Basic Properties of Trigonometric Functions 23 1.4. Solving the Simplest Trigonometric Equations. Inverse Trigonometric Functions 31 Problems 36 Chapter 2. Identical Transformations of Trigonometric Expressions 41 Inverse Trigonometric Functions Inverse Trigonometric Functions By: OpenStaxCollege For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle This is where the notion of an inverse to a trigonometric function comes into play In this section, we will explore the inverse trigonometric functions Understanding and Using the Inverse Sine, Cosine, and Tangent Functions In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in [link] For example, if f(x) = sin x, then we would write f − 1(x) = sin − 1x Be aware that sin − 1x does not mean sinx The following examples illustrate the inverse trigonometric functions: • Since sin ( π6 ) = 12 , then π ( 12 ) = sin − • Since cos(π) = − 1, then π = cos − 1( − 1) • Since tan ( π4 ) = 1, then π = tan − 1(1) 1/49 Inverse Trigonometric Functions In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value For this, we need inverse functions Recall that, for a one-to-one function, if f(a) = b, then an inverse function would satisfy f − 1(b) = a Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions The graph of each function would fail the horizontal line test In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one We choose a domain for each function that π π includes the number [link] shows the graph of the sine function limited to − , [ ] and the graph of the cosine function limited to [0, π] [ π (a) Sine function on a restricted domain of − , π ]; (b) Cosine function on a restricted domain of [0, π] ( π [link] shows the graph of the tangent function limited to − , π ) 2/49 Inverse Trigonometric Functions ( π Tangent function on a restricted domain of − , π ) These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote On these restricted domains, we can define the inverse trigonometric functions • The inverse sine function y = sin − 1x means x = sin y The inverse sine function is sometimes called the arcsine function, and notated arcsinx π π y = sin − 1x has domain [−1, 1] and range − , 2 • The inverse cosine function y = cos − 1x means x = cos y The inverse cosine function is sometimes called the arccosine function, and notated arccos x y = cos − 1x has domain [−1, 1] and range [0, π] • The inverse tangent function y = tan − 1x means x = tan y The inverse tangent function is sometimes called the arctangent function, and notated arctan x π π y = tan − 1x has domain (−∞, ∞) and range − , [ ] ( ) 3/49 Inverse Trigonometric Functions The graphs of the inverse functions are shown in [link], [link], and [link] Notice that the output of each of these inverse functions is a number, an angle in radian measure π π We see that sin − 1x has domain [−1, 1] and range − , , cos − 1x has domain [−1,1] [ ] ( π π ) and range [0, π], and tan − 1x has domain of all real numbers and range − , To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line y = x The sine function and inverse sine (or arcsine) function 4/49 Inverse Trigonometric Functions The cosine function and inverse cosine (or arccosine) function The tangent function and inverse tangent (or arctangent) function Relations for Inverse Sine, Cosine, and Tangent Functions [ π For angles in the interval − , π ], if sin y = x, then sin − 1x = y 5/49 Inverse Trigonometric Functions For angles in the interval [0, π], if cos y = x, then cos − 1x = y ( π For angles in the interval − , π ), if tan y = x, then tan − 1x = y Writing a Relation for an Inverse Function Given sin ( 5π12 ) ≈ 0.96593, write a relation involving the inverse sine Use the relation for the inverse ...CÁC HÀM LƯỢNG GIÁC - TRIGONOMETRIC FUNCTIONS ACOS (number) : Trả về một giá trị radian nằm trong khoảng từ 0 đến Pi, là arccosine, hay nghịch đảo cosine của một số nằm trong khoảng từ -1 đến 1 ACOSH (number) : Trả về một giá trị radian, là nghịch đảo cosine-hyperbol của một số lớn hơn hoặc bằng 1 ASIN (number) : Trả về một giá trị radian nằm trong đoạn từ -Pi/2 đến Pi/2, là arcsine, hay nghịch đảo sine của một số nằm trong khoảng từ -1 đến 1 ASINH (number) : Trả về một giá trị radian, là nghịch đảo sine-hyperbol của một số ATAN (number) : Trả về một giá trị radian nằm trong khoảng từ -Pi/2 đến Pi/2, là arctang, hay nghịch đảo tang của một số ATAN2 (x_num, y_num) : Trả về một giá trị radian nằm trong khoảng (nhưng không bao gồm) từ -Pi đến Pi, là arctang, hay nghịch đảo tang của một điểm có tọa độ x và y ATANH (number) : Trả về một giá trị radian, là nghịch đảo tang-hyperbol của một số nằm trong khoảng từ -1 đến 1 COS (number) : Trả về một giá trị radian, là cosine của một số COSH (number) : Trả về một giá trị radian, là cosine-hyperbol của một số DEGREES (angle) : Chuyển đổi số đo của một góc từ radian sang độ RADIANS (angle) : Chuyển đổi số đo của một góc từ độ sang radian SIN (number) : Trả về một giá trị radian là sine của một số SINH (number) : Trả về một giá trị radian, là sine-hyperbol của một số TAN (number) : Trả về một giá trị radian, là tang của một số TANH (number) : Trả về một giá trị radian, là tang-hyperbol của một số 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 Derivatives, Integrals, and Properties Of Inverse Trigonometric Functions and Hyperbolic Functions (On this handout, a represents a constant, u and x represent variable quantities) Derivatives of Inverse Trigonometric Functions Identities for Hyperbolic Functions d sin¡1 u dx sinh 2x = sinh x cosh x d cos¡1 u dx = p = p du ¡ u dx ¡1 du ¡ u2 dx d tan¡1 u = dx du + u2 dx d csc¡1 u dx = d sec¡1 u dx ¡1 du p juj u2 ¡ dx = d cot¡1 u dx = du p juj u2 ¡ dx (juj < 1) cosh 2x = cosh2 x + sinh2 x (juj < 1) Z Z p a2 a2 du ¡ u2 (juj > 1) ¡1 du + u2 dx du + u2 = sin¡1 = p du = u u ¡ a2 ³u´ +C a ³u´ tan¡1 +C a a ¯u¯ ¯ ¯ sec¡1 ¯ ¯ + C a a sinh x = ¡x x ¡x e ¡e sinh2 x = cosh 2x ¡ cosh x = e +e x = sinh x ex ¡ e¡x = x cosh x e + e¡x cschx = = x sinh x e ¡ e¡x sechx = = x cosh x e + e¡x coth x = cosh x ex + e¡x = x sinh x e ¡ e¡x cosh2 x ¡ sinh2 x = tanh2 x = ¡ sech2 x coth2 x = + csch2 x Derivatives of Hyperbolic Functions (Valid for u2 < a2 ) d sinh u dx = cosh u (Valid for all u) d cosh u dx = sinh u (Valid for u2 > a2 ) d u = dx The Six Basic Hyperbolic Functions x cosh 2x + (juj > 1) Integrals Involving Inverse Trigonometric Functions Z cosh2 x = du dx du dx sech2 u du dx du dx d coth u dx = ¡ csch2 u d sechu dx = ¡ sechu u d cschu dx = ¡ cschu coth u du dx du dx Inverse Hyperbolic Identities µ ¶ sech x = cosh x µ ¶ ¡1 ¡1 csch x = sinh x µ ¶ coth¡1 x = tanh¡1 x ¡1 ¡1 Integrals Involving Inverse Hyperbolic Functions Integrals of Hyperbolic Functions Z Z Z Z Z Z sinh u du Z = cosh u + C cosh u du Z = sinh u + C sech u du = u + C csch2 u du = ¡ coth u + C du a2 + u2 = sinh¡1 p du u2 ¡ a2 = cosh¡1 = ¡ cschu + C d cosh¡1 u dx = = d tanh¡1 u = dx d csch¡1 u dx = d sech¡1 u dx = d coth¡1 u dx = p du + u2 dx du p u ¡ dx (u > 1) du ¡ u2 dx (juj < 1) ¡1 du p juj + u dx (u 6= 0) ¡1 du p u ¡ u dx du ¡ u2 dx p du u § a2 p x2 + 1) cosh¡1 x = ln(x + p x2 ¡ 1) tanh¡1 x = du a ¡ u2 +C (a > 0) +C (u > a > 0) sech¡1 x csch¡1 x coth¡1 x = ln(u + p (¡1 < x < 1) (x ¸ 1) 1+x ln (jxj < 1) 1¡x à ! p + ¡ x2 = ln (0 < x · 1) x à ! p 1 + x2 = ln + (x 6= 0) x jxj = x+1 ln x¡1 u2 § a2 ) + C ¯ ¯ ¯a + u¯ ¯ ¯+C = ln 2a ¯ a ¡ u ¯ à ! p Z 1 a + a2 § u2 p du = ¡ ln +C a juj u a2 § u2 Z a = ln(x + Alternate Form For Integrals Involving Inverse Hyperbolic Functions Z ³u´ sinh¡1 x (0 < u < 1) (juj > 1) a Expressing Inverse Hyperbolic Functions As Natural Logarithms Derivatives of Inverse Hyperbolic Functions d sinh¡1 u dx ³u´ ³ ´ ¡1 u > > + C (if u2 < a2 ) > > Z a a < ³ ´ du = ¡1 u > a ¡ u2 > coth + C (if u2 > a2 ) > > a : a Z ³ ´ 1 ¡1 u p du = ¡ sech + C (0 < u < a) a a u a2 ¡ u2 Z ¯u¯ 1 ¯ ¯ p du = ¡ csch¡1 ¯ ¯ + C 2 a a u a +u sechu u du = ¡ sechu + C cschu coth u du p (jxj > 1) ... − , [ ] ( ) 3/49 Inverse Trigonometric Functions The graphs of the inverse functions are shown in [link], [link], and [link] Notice that the output of each of these inverse functions is a number,... with inverse trigonometric functions • Evaluate Expressions Involving Inverse Trigonometric Functions Visit this website for additional practice questions from Learningpod Key Concepts • An inverse. .. sine function and inverse sine (or arcsine) function 4/49 Inverse Trigonometric Functions The cosine function and inverse cosine (or arccosine) function The tangent function and inverse tangent