File0056 - 156 Chapter 3 Differentiation 41 y:xsin‘1x...

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Unformatted text preview: 156 Chapter 3 Differentiation 41. y:xsin‘1x+ l—X22xsin‘1x+(l—x2)l/2 _'—l x x __-— —sm X+m—m—Sln :> 3—: : sin'1x+x (7—11“) + G) (1 — x2)_1/2(—2x) 42. y : ln(x2 +4) — xtan‘1(§) => % = x2214 —-tan‘1(§)— x [—g] 2 172:4 —tan‘1(§)— flixx ) = — tan"1 ( Mix 43. The angle a is the large angle between the wall and the right end of the blackboard minus the small angle between the left end of the blackboard and the wall => a = cot‘1 (Ix—5) — cot‘1 G) . N 44. 65° + (90° — B) + (900 — a) 2 180° 2? oz = 65° — fl = 65° — tan‘1 ( 1) z 65° — 22.78° z 4222" “$1 45. Take each square as a unit square. From the diagram we have the following: the smallest angle a has a tangent of 1 => (1 = tan’1 1; the middle angle 6 has a tangent of 2 :> 6 2 tan‘1 2; and the largest angle 7 has a tangent of3 :> 'y = tan‘1 3. The sum of these three angles is 77 :> a + 6 + 'y = 71 => tan’11+tan‘12 + tan‘1 3 = 7r. 46. (a) From the symmetry of the diagram, we see that 7r — sec‘1 x is the vertical distance from the graph of l y : sec‘l x to the line y = 7r and this distance is the same as the height of y : sec‘ x above the x-axis at 1x = sec—1(-—x). 1x, where —1 g x 31: cos“1(— :> sec"1 (—x) 2 7r — sec‘1 x —x; i.e., 7r — sec” 1 (b) cos’1(—x) : 7r — cos‘ x) = 7r — cos’1 (i). where x 2 lorx S —l 47. Ifx =1:sin“(l)+ cos—1(1) = g + 0 : Ifx = 0: sin—1(0) + cos’1(0) : O + g = Ifx = —l:sin‘1(—1)+cos‘l(——1) = —§ +7r = g. The identity sin'1(x) + cos—1(x) = g has been established for x in (0, 1) , by Figure 1.6.7. So now ifx is in (~1, 0), note that —x is in (0, l), and we have that sin—1(x) + cos—1(x) : —sin“ (—x) + cos—1(x) since sin‘1 is odd 2 —sin‘1(—-x) + 7r — cos‘1 (—x) by Eq. 3, Section 1.6 = —(sin’1(—x)+ cos‘l(-x)) + 7r 2 _% + 7r NI: Nl=l ’ll' 2 This establishes the identity for all x in [—1, l]. 1 48. x:>tana:xandtanfi=%=>§=a+fl=tam x+tan’1%. 49. (a) Defined; there is an angle whose tangent is 2. (b) Not defined; there is no angle whose cosine is 2. 1 50. (a) Not defined; there is no angle whose cosecant is 5. (b) Defined; there is an angle whose cosecant is 2. ...
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