Ch 6.5: Inverse trigonometric functions

Transcription

1 Ch 6.5: Inverse trigonometric functions In this section, we will 1. define an one-to-one function 2. look at the conditions on which inverse functions exist 3. define and graph inverse functions for the six (6) trig. functions 4. look at common examples

2 One-to-One function A function f is one-to-one if every element in the range corresponds to only one element in the domain. That is, Definition (One-to-One function) A function f is one-to-one if x 1 x 2 f (x 1 ) f (x 2 ) Question) Are these functions 1-1?

3 Is the following function 1-1? Here s the graph of y = 1.

4 The Horizontal line test Recall that a function is one-to-one if for all x 1 and x 2, x 1 x 2 f (x 1 ) f (x 2 ). Thus, a function is not one-to-one if there are x 1 and x 2 with x 1 x 2 f (x 1 ) = f (x 2 ). That is, if there is a horizontal line that intersects the graph of the function two or more points, the function fails the horizontal line test and is not 1-1. Q: Which function(s) pass(es)/fail(s) the horizontal line test?

5 Why one-to-one? Given y = f (x), its inverse function, f 1, is when we solve for x and then interchange x and y. Thus, if a function f is 1-1 (i.e., if it passes the horizontal line test), the graph with x and y switched will pass the vertical line test, which will mean that f 1 exists as a function. (Note: All functions must pass the vertical line test.) Does the function pass the horizontal test? Does the inverse function exist?

6 Inverse of f (x) = x 2? Does the function pass the horizontal test? Does the inverse function exist? (If not, how can we make it exist?)

7 Graphs of inverse functions To find the graph of the inverse function, draw the (invisible) y = x line and flip everything about that line.

8 Graphs of inverse functions What about y = x 2 on [0, )?

9 A note about inverse functions Since we interchange x and y to find the inverse function of f the domain of f 1, (D(f 1 )), is the range of f, or (R(f )). the range of f 1, (R(f 1 )), is the domain of f, or (D(f )). Example) Given f = x 2 on [0, 2], find the domain and range of its inverse.

10 Inverse trig functions and their applications Question: Which of the sine, cosine and tangent functions pass(es) the horizontal test? That is, which of the three functions are 1-1?

11 Let s restrict the domain of the trig functions. Here s the sine function: How can we make it 1-1?

12 How about cosine? Here s the cosine function: How can we make it 1-1?

13 How about tangent? Here s the tangent function: How can we make it 1-1?

14 Summary- How to make these functions 1-1? For f (x) = sin x, restrict its domain to [ π 2, π 2 ]. For f (x) = cos x, restrict its domain to [0, π]. For f (x) = tan x, restrict its domain to ( π 2, π 2 ).

15 y = arcsin x = sin 1 (x) Here s the sine function restricted on [ π 2, π 2 ]. If we interchange x and y or flip it about the y = x, then

16 y = arcsin x = sin 1 (x) Domain: [ 1, 1] (= range of sin x) Range: [ π 2, π ] 2 ( = domain of sin x) sin(sin 1 x) = x for x sin 1 (sin x) = x for x

17 y = arccos x = cos 1 (x) Here s the cosine function restricted on [0, π]. If we interchange x and y or flip it about the y = x, we have

18 y = arccos x = cos 1 (x) Domain: [ 1, 1] (= range of cos x) Range: [0, π] ( = domain of cos x) cos(cos 1 x) = x for x cos 1 (cos x) = x for x

19 y = arctan x = tan 1 (x) Domain: (= range of tan x) Range: ( = domain of tan x) tan(tan 1 x) = x for tan 1 (tan x) = x for

20 Example 1 Solve each equation ( for y without the use of a calculator. 1. y = sin 1 ) y = arcsin ( 1 ) 2 3. y = sin 1 (2)

21

22 Example 2 Evaluate each expression: 1. y = sin [ sin 1 ( )] y = arcsin [ sin ( )] 3π 4 3. y = sin 1 [ sin ( )] 5π 6

23

24

25 Example 3 Evaluate the inverse cosine function for the values given: 1. y = cos 1 (0) ( 2. y = arccos 3. y = cos 1 (π) 3 2 )

26

27 Example 4 Evaluate each expression: 1. y = cos [ cos 1 ( 0.2) ] 2. y = arccos [ cos ( )] π y = cos 1 [ cos ( )] 4π 3

28

29 Example 5 Evaluate each expression: 1. y = tan 1 ( 3) 2. y = arctan [tan (.89)] 3. y = tan 1 [ tan ( )] 7π 6

30

31 Example 6 Simplify each expression: 1. y = tan [ sin 1 ( 1 )] 2 2. y = sin 1 [ cos ( )] π 3

32

33 Example 7 Evaluate the expression tan [ sin 1 ( 4 5)].

34 Example 8 Evaluate the following expression as an algebraic function of x: sin [ cos 1 2x ]. Assume 1 2 x 1 2.

35 Example 9 ( Evaluate the expression tan [cos 1 x x )]. Assume x > 0.

36 Inverse functions for secant (Not covered in textbook) First, here s the graph of the secant function. then, we may restrict our domain to [0, π 2 ) ( π 2, π], so that y = sec 1 x can be defined.

37 Graph of y = sec 1 x Insert graph of the inverse function here. Domain Range

38 Inverse functions for cosecant First, here s the graph of the cosecant function. then, we may restrict our domain to [ π 2, 0) (0, π 2 ], so that y = csc 1 x can be defined.

39 graph of y = csc 1 x Domain Range

40 Inverse functions for cotangent First, here s the graph of the cotangent function. then, we may restrict our domain to either [ π 2, 0) (0, π 2 ) or (0, π), so that the graph of the y = cot 1 x is then

41 y = cot 1 x or Domain Range

42 Example 10 Evaluate the following: 1. csc 1 (2) ( ) 2. sec ( ) 3. cot 1 3 1

43