Section	3.5
     Inverse	Trigonometric
           Functions
               V63.0121.006/016, Calculus	I



               ...
Announcements




     Exams	returned	in	recitation
     There	is	WebAssign	due	Tuesday	March	23	and	written	HW
     due	T...
What	is	an	inverse	function?



   Definition
   Let f be	a	function	with	domain D and	range E. The inverse of f is
   the	...
What	is	an	inverse	function?



   Definition
   Let f be	a	function	with	domain D and	range E. The inverse of f is
   the	...
What	functions	are	invertible?



   In	order	for f−1 to	be	a	function, there	must	be	only	one a in D
   corresponding	to	...
Outline


  Inverse	Trigonometric	Functions


  Derivatives	of	Inverse	Trigonometric	Functions
     Arcsine
     Arccosine...
arcsin
   Arcsin	is	the	inverse	of	the	sine	function	after	restriction	to
   [−π/2, π/2].

                               ...
arcsin
   Arcsin	is	the	inverse	of	the	sine	function	after	restriction	to
   [−π/2, π/2].

                               ...
arcsin
   Arcsin	is	the	inverse	of	the	sine	function	after	restriction	to
   [−π/2, π/2].

                               ...
arcsin
   Arcsin	is	the	inverse	of	the	sine	function	after	restriction	to
   [−π/2, π/2].

                               ...
arccos
   Arccos	is	the	inverse	of	the	cosine	function	after	restriction	to
   [0, π]



                                 ...
arccos
   Arccos	is	the	inverse	of	the	cosine	function	after	restriction	to
   [0, π]



                                 ...
arccos
   Arccos	is	the	inverse	of	the	cosine	function	after	restriction	to
   [0, π]



                                 ...
arccos
   Arccos	is	the	inverse	of	the	cosine	function	after	restriction	to
   [0, π]

                                . ....
arctan
   Arctan	is	the	inverse	of	the	tangent	function	after	restriction	to
   [−π/2, π/2].
                             ...
arctan
   Arctan	is	the	inverse	of	the	tangent	function	after	restriction	to
   [−π/2, π/2].
                             ...
arctan
   Arctan	is	the	inverse	of	the	tangent	function	after	restriction	to
                                             ...
arctan
   Arctan	is	the	inverse	of	the	tangent	function	after	restriction	to
   [−π/2, π/2].
                             ...
arcsec
   Arcsecant	is	the	inverse	of	secant	after	restriction	to
   [0, π/2) ∪ (π, 3π/2].
                               ...
arcsec
   Arcsecant	is	the	inverse	of	secant	after	restriction	to
   [0, π/2) ∪ (π, 3π/2].
                               ...
arcsec
   Arcsecant	is	the	inverse	of	secant	after	restriction	to
                                                        ...
arcsec                           3π
                                 .
   Arcsecant	is	the	inverse	of	secant	after	restric...
Values	of	Trigonometric	Functions

                  π        π        π                 π
       x 0
                  6 ...
Check: Values	of	inverse	trigonometric	functions

   Example
   Find
          arcsin(1/2)
          arctan(−1)
          ...
Check: Values	of	inverse	trigonometric	functions

   Example
   Find
          arcsin(1/2)
          arctan(−1)
          ...
What	is arctan(−1)?

                 .


    3
    . π/4
            .




                 .                .




      ...
What	is arctan(−1)?

                          .

                                                          (        )
   ...
What	is arctan(−1)?

                          .

                                                          ( )
    3
    ...
What	is arctan(−1)?

                    .

                                                          (   )
    3
    . π/...
What	is arctan(−1)?

                    .

                                                          (   )
    3
    . π/...
Check: Values	of	inverse	trigonometric	functions

   Example
   Find
          arcsin(1/2)
          arctan(−1)
          ...
Check: Values	of	inverse	trigonometric	functions

   Example
   Find
          arcsin(1/2)
          arctan(−1)
          ...
Caution: Notational	ambiguity




           . in2 x =.(sin x)2
           s                             . in−1 x = (sin x...
Outline


  Inverse	Trigonometric	Functions


  Derivatives	of	Inverse	Trigonometric	Functions
     Arcsine
     Arccosine...
Theorem	(The	Inverse	Function	Theorem)
Let f be	differentiable	at a, and f′ (a) ̸= 0. Then f−1 is	defined	in	an
open	interv...
Theorem	(The	Inverse	Function	Theorem)
Let f be	differentiable	at a, and f′ (a) ̸= 0. Then f−1 is	defined	in	an
open	interv...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                    dy        dy     1          1
     ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
The	derivative	of	arcsin

   Let y = arcsin x, so x = sin y. Then

                     dy        dy     1          1
    ...
Graphing	arcsin	and	its	derivative


                                                                 1
                  ...
The	derivative	of	arccos

   Let y = arccos x, so x = cos y. Then

                   dy        dy      1             1
  ...
The	derivative	of	arccos

   Let y = arccos x, so x = cos y. Then

                    dy        dy      1             1
 ...
Graphing	arcsin	and	arccos



      . . rccos
        a


                   . . rcsin
                     a


       .
 ...
Graphing	arcsin	and	arccos



      . . rccos
        a
                               Note
                              ...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                    dy        dy     1
           sec2 ...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                     dy        dy     1
            sec...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                     dy        dy     1
            sec...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                     dy        dy     1
            sec...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                     dy        dy     1
            sec...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                     dy        dy     1
            sec...
The	derivative	of	arctan

   Let y = arctan x, so x = tan y. Then

                      dy        dy     1
             s...
Graphing	arctan	and	its	derivative


                              y
                              .
                     ...
Example
                    √
Let f(x) = arctan    x. Find f′ (x).




                                       .   .   .   ...
Example
                    √
Let f(x) = arctan    x. Find f′ (x).

Solution

         d        √       1     d√     1   1...
The	derivative	of	arcsec

   Try	this	first.




                           .   .   .   .   .   .
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                    dy        dy        ...
The	derivative	of	arcsec

   Try	this	first. Let y = arcsec x, so x = sec y. Then

                      dy        dy      ...
Another	Example




  Example
  Let f(x) = earcsec x . Find f′ (x).




                                        .   .   . ...
Another	Example




  Example
  Let f(x) = earcsec x . Find f′ (x).

  Solution
                                          ...
Outline


  Inverse	Trigonometric	Functions


  Derivatives	of	Inverse	Trigonometric	Functions
     Arcsine
     Arccosine...
Application


  Example
  One	of	the	guiding	principles
  of	most	sports	is	to	“keep
  your	eye	on	the	ball.” In
  basebal...
Let y(t) be	the	distance	from	the	ball	to	home	plate, and θ the
angle	the	batter’s	eyes	make	with	home	plate	while	followi...
Let y(t) be	the	distance	from	the	ball	to	home	plate, and θ the
 angle	the	batter’s	eyes	make	with	home	plate	while	follow...
Let y(t) be	the	distance	from	the	ball	to	home	plate, and θ the
 angle	the	batter’s	eyes	make	with	home	plate	while	follow...
Let y(t) be	the	distance	from	the	ball	to	home	plate, and θ the
 angle	the	batter’s	eyes	make	with	home	plate	while	follow...
Recap

        y         y′

                   1
    arcsin x   √
                 1 − x2
                    1
    arcco...
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Lesson 15: Inverse Trigonometric Functions

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Lesson 15: Inverse Trigonometric Functions

  1. 1. Section 3.5 Inverse Trigonometric Functions V63.0121.006/016, Calculus I March 11, 2010 Announcements Exams returned in recitation There is WebAssign due Tuesday March 23 and written HW due Thursday March 25 . . . . . .
  2. 2. Announcements Exams returned in recitation There is WebAssign due Tuesday March 23 and written HW due Thursday March 25 next quiz is Friday April 2 . . . . . .
  3. 3. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. . . . . . .
  4. 4. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. So f−1 (f(x)) = x, f(f−1 (x)) = x . . . . . .
  5. 5. What functions are invertible? In order for f−1 to be a function, there must be only one a in D corresponding to each b in E. Such a function is called one-to-one The graph of such a function passes the horizontal line test: any horizontal line intersects the graph in exactly one point if at all. If f is continuous, then f−1 is continuous. . . . . . .
  6. 6. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
  7. 7. arcsin Arcsin is the inverse of the sine function after restriction to [−π/2, π/2]. y . . . . x . π π s . in − . . 2 2 . . . . . .
  8. 8. arcsin Arcsin is the inverse of the sine function after restriction to [−π/2, π/2]. y . . . . . x . π π s . in − . . . 2 2 . . . . . .
  9. 9. arcsin Arcsin is the inverse of the sine function after restriction to [−π/2, π/2]. y . y . =x . . . . x . π π s . in − . . . 2 2 . . . . . .
  10. 10. arcsin Arcsin is the inverse of the sine function after restriction to [−π/2, π/2]. y . . . rcsin a . . . . x . π π s . in − . . . 2 2 . The domain of arcsin is [−1, 1] [ π π] The range of arcsin is − , 2 2 . . . . . .
  11. 11. arccos Arccos is the inverse of the cosine function after restriction to [0, π] y . c . os . . x . 0 . . π . . . . . .
  12. 12. arccos Arccos is the inverse of the cosine function after restriction to [0, π] y . . c . os . . x . 0 . . π . . . . . . .
  13. 13. arccos Arccos is the inverse of the cosine function after restriction to [0, π] y . y . =x . c . os . . x . 0 . . π . . . . . . .
  14. 14. arccos Arccos is the inverse of the cosine function after restriction to [0, π] . . rccos a y . . c . os . . . x . 0 . . π . The domain of arccos is [−1, 1] The range of arccos is [0, π] . . . . . .
  15. 15. arctan Arctan is the inverse of the tangent function after restriction to [−π/2, π/2]. y . . x . 3π π π 3π − . − . . . 2 2 2 2 t .an . . . . . .
  16. 16. arctan Arctan is the inverse of the tangent function after restriction to [−π/2, π/2]. y . . x . 3π π π 3π − . − . . . 2 2 2 2 t .an . . . . . .
  17. 17. arctan Arctan is the inverse of the tangent function after restriction to y . =x [−π/2, π/2]. y . . x . 3π π π 3π − . − . . . 2 2 2 2 t .an . . . . . .
  18. 18. arctan Arctan is the inverse of the tangent function after restriction to [−π/2, π/2]. y . π . a . rctan 2 . x . π − . 2 The domain of arctan is (−∞, ∞) ( π π) The range of arctan is − , 2 2 π π lim arctan x = , lim arctan x = − x→∞ 2 x→−∞ 2 . . . . . .
  19. 19. arcsec Arcsecant is the inverse of secant after restriction to [0, π/2) ∪ (π, 3π/2]. y . . x . 3π π π 3π − . − . . . 2 2 2 2 s . ec . . . . . .
  20. 20. arcsec Arcsecant is the inverse of secant after restriction to [0, π/2) ∪ (π, 3π/2]. y . . . x . 3π π π 3π − . − . . . . 2 2 2 2 s . ec . . . . . .
  21. 21. arcsec Arcsecant is the inverse of secant after restriction to y . =x [0, π/2) ∪ (π, 3π/2]. y . . . x . 3π π π 3π − . − . . . . 2 2 2 2 s . ec . . . . . .
  22. 22. arcsec 3π . Arcsecant is the inverse of secant after restriction to 2 [0, π/2) ∪ (π, 3π/2]. . . y π . 2 . . . x . . The domain of arcsec is (−∞, −1] ∪ [1, ∞) [ π ) (π ] The range of arcsec is 0, ∪ ,π 2 2 π 3π lim arcsec x = , lim arcsec x = x→∞ 2 x→−∞ 2 . . . . . .
  23. 23. Values of Trigonometric Functions π π π π x 0 6 4 3 2 √ √ 1 2 3 sin x 0 1 2 2 2 √ √ 3 2 1 cos x 1 0 2 2 2 1 √ tan x 0 √ 1 3 undef 3 √ 1 cot x undef 3 1 √ 0 3 2 2 sec x 1 √ √ 2 undef 3 2 2 2 csc x undef 2 √ √ 1 2 3 . . . . . .
  24. 24. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(−1) ( √ ) 2 arccos − 2 . . . . . .
  25. 25. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(−1) ( √ ) 2 arccos − 2 Solution π 6 . . . . . .
  26. 26. What is arctan(−1)? . 3 . π/4 . . . . − . π/4 . . . . . .
  27. 27. What is arctan(−1)? . ( ) 3 . π/4 3π . Yes, tan = −1 4 √ 2 s . in(3π/4) = 2 . √ . 2 . os(3π/4) = − c 2 . − . π/4 . . . . . .
  28. 28. What is arctan(−1)? . ( ) 3 . π/4 3π . Yes, tan = −1 4 √ But, the range of arctan ( π π) 2 s . in(3π/4) = is − , 2 2 2 . √ . 2 . os(3π/4) = − c 2 . − . π/4 . . . . . .
  29. 29. What is arctan(−1)? . ( ) 3 . π/4 3π . Yes, tan = −1 4 But, the range of arctan ( π π) √ is − , 2 2 2 c . os(π/4) = . 2 Another angle whose . π tangent is −1 is − , and √ 4 2 this is in the right range. . in(π/4) = − s 2 . − . π/4 . . . . . .
  30. 30. What is arctan(−1)? . ( ) 3 . π/4 3π . Yes, tan = −1 4 But, the range of arctan ( π π) √ is − , 2 2 2 c . os(π/4) = . 2 Another angle whose . π tangent is −1 is − , and √ 4 2 this is in the right range. . in(π/4) = − s π 2 So arctan(−1) = − 4 . − . π/4 . . . . . .
  31. 31. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(−1) ( √ ) 2 arccos − 2 Solution π 6 π − 4 . . . . . .
  32. 32. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(−1) ( √ ) 2 arccos − 2 Solution π 6 π − 4 3π 4 . . . . . .
  33. 33. Caution: Notational ambiguity . in2 x =.(sin x)2 s . in−1 x = (sin x)−1 s sinn x means the nth power of sin x, except when n = −1! The book uses sin−1 x for the inverse of sin x, and never for (sin x)−1 . 1 I use csc x for and arcsin x for the inverse of sin x. sin x . . . . . .
  34. 34. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
  35. 35. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) . . . . . .
  36. 36. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) “Proof”. If y = f−1 (x), then f(y ) = x , So by implicit differentiation dy dy 1 1 f′ (y) = 1 =⇒ = ′ = ′ −1 dx dx f (y) f (f (x)) . . . . . .
  37. 37. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) . . . . . .
  38. 38. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: . . . . . . .
  39. 39. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . . . . . . . .
  40. 40. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . y . = arcsin x . . . . . . .
  41. 41. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . y . = arcsin x . √ . 1 − x2 . . . . . .
  42. 42. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: √ cos(arcsin x) = 1 − x2 1 . x . y . = arcsin x . √ . 1 − x2 . . . . . .
  43. 43. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: √ cos(arcsin x) = 1 − x2 1 . x . So d 1 y . = arcsin x arcsin(x) = √ dx 1 − x2 . √ . 1 − x2 . . . . . .
  44. 44. Graphing arcsin and its derivative 1 .√ 1 − x2 The domain of f is [−1, 1], but the domain . . rcsin a of f′ is (−1, 1) lim f′ (x) = +∞ x →1 − lim f′ (x) = +∞ . | . . | x→−1+ − . 1 1 . . . . . . . .
  45. 45. The derivative of arccos Let y = arccos x, so x = cos y. Then dy dy 1 1 − sin y = 1 =⇒ = = dx dx − sin y − sin(arccos x) . . . . . .
  46. 46. The derivative of arccos Let y = arccos x, so x = cos y. Then dy dy 1 1 − sin y = 1 =⇒ = = dx dx − sin y − sin(arccos x) To simplify, look at a right triangle: √ sin(arccos x) = 1 − x2 1 . √ . 1 − x2 So d 1 y . = arccos x arccos(x) = − √ . dx 1 − x2 x . . . . . . .
  47. 47. Graphing arcsin and arccos . . rccos a . . rcsin a . | . |. . − . 1 1 . . . . . . . .
  48. 48. Graphing arcsin and arccos . . rccos a Note (π ) cos θ = sin −θ . . rcsin a 2 π =⇒ arccos x = − arcsin x 2 . | . |. . So it’s not a surprise that their − . 1 1 . derivatives are opposites. . . . . . . .
  49. 49. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y . . . . . .
  50. 50. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: . . . . . . .
  51. 51. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: x . . 1 . . . . . . .
  52. 52. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: x . y . = arctan x . 1 . . . . . . .
  53. 53. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: √ . 1 + x2 x . y . = arctan x . 1 . . . . . . .
  54. 54. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: 1 cos(arctan x) = √ 1 + x2 √ . 1 + x2 x . y . = arctan x . 1 . . . . . . .
  55. 55. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: 1 cos(arctan x) = √ 1 + x2 √ . 1 + x2 x . So d 1 y . = arctan x arctan(x) = . dx 1 + x2 1 . . . . . . .
  56. 56. Graphing arctan and its derivative y . . /2 π a . rctan 1 . 1 + x2 . x . − . π/2 The domain of f and f′ are both (−∞, ∞) Because of the horizontal asymptotes, lim f′ (x) = 0 x→±∞ . . . . . .
  57. 57. Example √ Let f(x) = arctan x. Find f′ (x). . . . . . .
  58. 58. Example √ Let f(x) = arctan x. Find f′ (x). Solution d √ 1 d√ 1 1 arctan x = (√ )2 x= · √ dx 1+ x dx 1+x 2 x 1 = √ √ 2 x + 2x x . . . . . .
  59. 59. The derivative of arcsec Try this first. . . . . . .
  60. 60. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) . . . . . .
  61. 61. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: . . . . . . .
  62. 62. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: . . . . . . .
  63. 63. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: x . . 1 . . . . . . .
  64. 64. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: x . y . = arcsec x . 1 . . . . . . .
  65. 65. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: √ x2 − 1 tan(arcsec x) = √ 1 x . . x2 − 1 y . = arcsec x . 1 . . . . . . .
  66. 66. The derivative of arcsec Try this first. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: √ x2 − 1 tan(arcsec x) = √ 1 x . . x2 − 1 So d 1 y . = arcsec x arcsec(x) = √ . dx x x2 − 1 1 . . . . . . .
  67. 67. Another Example Example Let f(x) = earcsec x . Find f′ (x). . . . . . .
  68. 68. Another Example Example Let f(x) = earcsec x . Find f′ (x). Solution 1 f′ (x) = earcsec x · √ x x2 − 1 . . . . . .
  69. 69. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
  70. 70. Application Example One of the guiding principles of most sports is to “keep your eye on the ball.” In baseball, a batter stands 2 ft away from home plate as a pitch is thrown with a velocity of 130 ft/sec (about 90 mph). At what rate does the batter’s angle of gaze need to change to follow the ball as it crosses home plate? . . . . . .
  71. 71. Let y(t) be the distance from the ball to home plate, and θ the angle the batter’s eyes make with home plate while following the ball. We know y′ = −130 and we want θ′ at the moment that y = 0. y . 1 . 30 ft/sec . θ . . 2 . ft . . . . . .
  72. 72. Let y(t) be the distance from the ball to home plate, and θ the angle the batter’s eyes make with home plate while following the ball. We know y′ = −130 and we want θ′ at the moment that y = 0. We have θ = arctan(y/2). Thus dθ 1 1 dy = · 2 2 dt dt 1 + ( y /2 ) y . 1 . 30 ft/sec . θ . . 2 . ft . . . . . .
  73. 73. Let y(t) be the distance from the ball to home plate, and θ the angle the batter’s eyes make with home plate while following the ball. We know y′ = −130 and we want θ′ at the moment that y = 0. We have θ = arctan(y/2). Thus dθ 1 1 dy = · 2 2 dt dt 1 + ( y /2 ) When y = 0 and y′ = −130, y . then dθ 1 1 = · (−130) = −65 rad/sec 1 . 30 ft/sec dt y =0 1+0 2 . θ . . 2 . ft . . . . . .
  74. 74. Let y(t) be the distance from the ball to home plate, and θ the angle the batter’s eyes make with home plate while following the ball. We know y′ = −130 and we want θ′ at the moment that y = 0. We have θ = arctan(y/2). Thus dθ 1 1 dy = · 2 2 dt dt 1 + ( y /2 ) When y = 0 and y′ = −130, y . then dθ 1 1 = · (−130) = −65 rad/sec 1 . 30 ft/sec dt y =0 1+0 2 . θ The human eye can only . track at 3 rad/sec! . 2 . ft . . . . . .
  75. 75. Recap y y′ 1 arcsin x √ 1 − x2 1 arccos x − √ Remarkable that the 1 − x2 derivatives of these 1 transcendental functions arctan x 1 + x2 are algebraic (or even 1 rational!) arccot x − 1 + x2 1 arcsec x √ x x2 − 1 1 arccsc x − √ x x2 − 1 . . . . . .

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