## Asadkarim7 5 years ago the tangent and normal to the curve y= sin x at the point P where x= π/3 cut the x-axis at A and B respectively (a). show that AB = (5√3)/4. (b). find the area of the triangle PAB.

1. amistre64

the tangent to the curve is what..cos(pi/3) right?

2. amistre64

the normal is 90 from that....

3. amistre64

i dont quite understand the cuts the x axis at A and B

4. anonymous

maybe x and y axis at A and B?

5. amistre64

6. anonymous

no

7. anonymous

they mean the tangent cuts at a, and the normal cuts at b

let me post the digaram

9. anonymous

its very simple find the eqns of the lines , find the intercepts of those lines , then use distance formula

10. amistre64

lol.... yeah, if you post it that would be more helpful :)

11. amistre64

simple yes, once you understand the question ;)

12. anonymous

just distance formula and area of a triangle = (1/2) bh , thats all there is , you dont even need a picture for this, should be able to draw one yourself, or visualise in your mind pretty easily

14. amistre64

well the tangent is simply its derivative; cos(pi/3)

15. amistre64

the equation of the line then becomes: y-(sin(pi/3)) = cos(pi/3)(x-(pi/3))

16. amistre64

cos(pi/3) = 1/2 right?

17. amistre64

the normal will have a slope of -2 then

18. amistre64

y-(sin(pi/3)) = -2(x-(pi/3)) for the normal equation

Guys i still cant understand it , can sun1 explain it to me step by step ? plz

20. amistre64

they give you a point (pi/3, sin(pi/3)); or rather (pi/3, sqrt(3)/2) right?

21. anonymous

first take the derivative of sine, which is cosine. then plug in $\frac{\pi}{3}$ into the derivative to get the slope. $cos(\frac{\pi}{3})=\frac{1}{2}$ so that is the slope of the tangent line

22. amistre64

at this level you should have some redimentary skills at a graph id assume :)

23. amistre64

use the poit-slope form of a line to get the euation for the tangent and the norml

24. anonymous

amistre is in right direction,find the eqn of tangent and normal to the curve, find the x intercepts of both to find out the coordiantes of the two points, point P is given, it forms the triangle ABP

25. amistre64

someones been rearragnging my keyboard lol

26. anonymous

the equation of the line use the point slope formula. the point is $(\frac{\pi}{3},\frac{\sqrt{3}}{2})$ and the slope is $\frac{1}{2}$ so the equation is $y-\frac{\sqrt{3}}{2}=\frac{1}{2}(x-\frac{\pi}{3})$

27. anonymous

this crosses the x axis where $y=0$ so set $y=0$ and solve to get $x=\frac{\pi}{3}-\sqrt{3}$ if i did the algebra right

28. anonymous

repeat the process for the line with slope -2 through $(\frac{\pi}{3},\frac{\sqrt{3}}{2})$

29. amistre64

x/2 - pi/6 = -sqrt.3/2 x = -sqrt3 + pi/3 its good ;)

30. anonymous

amistre did i screw this up? it looks good to me

31. anonymous

thnx

32. amistre64

the normal just uses -2 for th eslope then

33. anonymous

$y-\frac{\sqrt{3}}{2}=-2(x-\frac{\pi}{3})$ is the normal line yes?

34. amistre64

yes

35. anonymous

let $y=0$ get $x=\sqrt{3}+\frac{\pi}{3}$

36. anonymous

at least that is what i got

37. amistre64

/ -2 to begin with to ease the process

38. amistre64

sqrt.3/4 + pi/3 i think

39. anonymous

$-\frac{\sqrt{3}}{2}=-2(x-\frac{\pi}{3})$ $\sqrt{3}=x-\frac{\pi}{3}$ $x=\sqrt{3}+\frac{\pi}{3}$

40. anonymous

whoooooooooooooooooops

41. anonymous

lorda mercy

42. anonymous

$\frac{\sqrt{3}}{4}+\frac{\pi}{3}$

43. anonymous

another dumb guy award for me.

44. anonymous

thnx

45. amistre64

lol.... thats what all my medals are for ;)

46. anonymous

ok fine. now that we have the x-intercepts, after some elementary algebra mistakes on my part, what are we supposed to do with them?

47. amistre64

throw tham at moving trains :)

48. anonymous

u both deserve for it :)

49. anonymous

i forgot what we were supposed to do with these. multiply? add? find the length?

50. anonymous

i mean for medals ;)..not throwing at the moving train :P

51. amistre64

the distance between them is just large - small

52. anonymous

now the final task, area of the triangle

53. amistre64

since they are on the number line; distance between them is large - small

54. amistre64

distance between 6 and -3 is 6--3 = 9 ;)

55. amistre64

height = sqrt3/2 ; distance = l-s --------------------------- = area 2

56. anonymous

meant to find an area lol but you guys have done it all over the place and its highly likely that the OP has just giving up on trying to follow what you guys have wrote

57. amistre64

the journey taken is more important the the final destination....... grasshopper

58. amistre64

sats been consistently and systematically finding the points... its good

59. anonymous

ooh some arithmetic i can do! $\frac{\sqrt{3}}{4}+\frac{\pi}{3}-(\frac{\pi}{3}-\sqrt{3})=\frac{\sqrt{3}}{4}+\sqrt{3}=\frac{5\sqrt{3}}{4}$

60. anonymous

wow just what it said it was!

61. anonymous

very east to set out working out to make the solution of these questions look really easy instead of making it look so complicated with 1000s of posts etc.

62. anonymous

true but very hard to type all in in one box. specially if typesetting.

63. amistre64

depends on if you wanna read a novel or have a conversation type environment; there isnt only one way to do things :)

64. anonymous

hi yo man

65. amistre64

this type of environment is suited best for interaction between questioneer and answeer

66. amistre64

if they really want a static read thru answer; theres always the internet lol

67. anonymous

Asadkarim7 i will be happy to work all this out on one paper and send it to you if this is confusing. the idea is this: find the slope of the tangent line, then find its equation. find the slope of the normal line, and its equation. then in each equation set y = 0 and solve for x. that is the whole idea. once you have both x intercepts subtract to get the distance. the rest is mechanics, but i will be happy to write it out in all its gory details without my algebra errors if you like

68. amistre64

69. anonymous

i am sure he is but i will be happy to do it to practice my latex

70. anonymous

make a pdf

71. amistre64

:) latex.... i perfer ascii lol

thanks :P amistre64

73. anonymous

luddite

74. anonymous

^ "its very simple find the eqns of the lines , find the intercepts of those lines , then use distance formula" esessentially what I wrote like a whole hour ago

75. amistre64

ohhh..... so everything after that is useless and should be erased becasue its just utter garbage? .... comes across a bit pompuos dontcha think ;)

76. amistre64

in fact the "i still dont understand can you explain it to me step by step" kinda leads me to believe otherwise ... lol

77. anonymous

also, geez, way to spam the question guys, nw my computer is going extremely slow when I view this particular question because you guys spammed it so much :|

78. amistre64

youre prolly using internet explorer; i had to down firefox to avoid that

79. amistre64

the equation editor stuff tends to put a strain on the processor; which is why i tend to avoid it

80. amistre64

81. anonymous

here it is

82. anonymous

let me know if you have any questions

83. amistre64

you forgot to put in the area lol

84. anonymous

oh heavens. fine.

85. amistre64

and here my fine useless garbage contribution ;)

sattelite73 u made the equation y = sqrt 3/ 2 it should have been y- sqrt 3/2

87. anonymous

$\frac{1}{2}bh=\frac{1}{2}\times \frac{5\sqrt{3}}{4}\times \frac{\sqrt{3}}{2}=\frac{15}{16}$

88. amistre64

its his latex.... makes him go a little insane at times and starts putting = signs for - signs :)

89. anonymous

everyone is a critic. ok that was a typo and look, it is gone!

90. amistre64

that ones just blank.....

u r genius sattelite 73

92. anonymous

what lovely font! what nice equations!

93. anonymous

how didi u make pdf file

94. anonymous

eventually i will learn this latex ... eventually.

95. amistre64

he starts out with some fine kindling, then adds a bit of tinder to feed the flame..

96. anonymous

i am using texmaker as an editor. just converts for you if you ask, and then can ask for external viewer. really quite friendly. i tried to use this on micordemon windows whatever and it took me maybe a day and a half to download what i needed. miktex and all that garbage. never worked right. left the darkside for linux a few months ago, to maybe 2 minutes do download and works like a charm.

97. anonymous

it is true i start with the trees. used to roll my own when i smoked. also doing html by hand if i can (with editor of course)

98. amistre64

<html> <body> <input type=button value="Click me!!" onClick="qwe()"> </body> <script language=javascript> function qwe() {alert("Quit it!")} </script> </html>

99. anonymous

also spelled "point" and 'derivative' wrong. perils of not using wysiwyg

100. anonymous

armistre will this make my computer explode?

101. amistre64

102. amistre64

hopefully lol

103. anonymous

this will keep me busy for at least an hour. click, click, ...

104. amistre64

lol :)

Thanks Everyone for your contributions , and giving your precious time to my question <3

106. amistre64

i made a 'flip book' javascript movie maker for my neices

107. amistre64

youre welcome :) and thank you elec for making it all possible lol

108. anonymous

welcome. hope it is clear enough even with all this garbage thrown in.