INVERSE-TRIGONOMETRIC-FUNCTIONS：反三角函数课件.ppt

1、4.7 INVERSE TRIGONOMETRIC FUNCTIONSFor an inverse to exist the function MUST be one-to-one A function is one-to-one if for every x there is exactly one y and for every y there is exactly one x.So If x and/or y is raised to an even power then the inverse does not exist unless the domain is restricted

2、.The equation y=x2 does not have an inverse because two different x values will produce the same y-value.i.e.x=2 and x=-2 will produce y=4.The horizontal line test fails.In order to restrict the domain,a basic knowledge of the shape of the graph is crucial.This is a parabola with(0,0)as the vertex.R

3、estrict the domain to the interval 0,infinity)to make it one-to-one.Now lets look at the trig functionsxyxyxyy=sin xy=cos xy=tan xxyFor the graph of y=sin x,the Domain is(-,)the Range is -1,1Not a 1-1 functionSo it currently does not have an inversexyHowever we can restrict the domain to-/,/Note the

4、 range will remain -1,1Now its 1-1!xyy=sinxThe inverse of sinxor Is denoted as arcsinxx1sinOn the unit circle:xyFor the inverse sine function with angles only from-/to /our answers will only be in either quadrant 1 for positive values and quadrant 4 for negative values.Find the exact value,if possib

5、le,1-113arcsin sin sin 222xyy=cos x is not one to one,so its domain will also need to be restricted.y=cos x is not one to one,so its domain will also need to be restricted./xyOn this interval,0,the cosine function is one-to-one and we can now define the inverse cosine function.y=arccos x or y=cos-1

6、xy=arccos x y=cos x On the unit circle,inverse cosine will only exist in quadrant 1 if the value is positive and quadrant 2 if the value is negative.xyFind the exact value for:-123arccos arccos(1)cos22y=tan x/xyRemember that tangent is undefined at-/and /y=tanx y=arctanx/xyRemember that tangent is undefined at-/and /Find the exact value13arctan1 tan0 arctan3Using the calculator.Be in radian mode Arctan(-15.7896)Arcsin(.3456)Arccos(-.6897)Arcsin(1.4535)Arccos(-2.4534)H Dub 4-7 Page 349#1-16all,49-67odd