Section 5.4 - Inverse Trigonometry RECALL – Facts About .

Section 5.4 - Inverse TrigonometryIn this section, we will define inverse since, cosine and tangent functions.RECALL – Facts about inverse functions:A function f ( x) is one-to-one if no two different inputs produce the same output(or: passes the horizontal line test)Example: f ( x) x 2 is NOT one-to-one. g ( x) x 3 is one-to-one.A function f ( x) is invertible if it is one-to-one.The inverse function is notated as: f 1 ( x) .f : A Bf 1 : B ADomain: ADomain: BRange: BRange: AImportant: f (a ) b if and only if f 1 (b) a .1

INVERSE SINE FUNCTIONHere’s the graph of f ( x) sin( x) .Domain: , Range: 1,1 Is it one to one?If the function is not one-to-one, we run into problems when we consider theinverse of the function. What we want to do with the sine function is to restrict thevalues for sine. When we make a careful restriction, we can get something that ISone-to-one.2

If we limit the function to the interval , , the graph will look like this: 2 2 Restricted Sine function Domain: , 2 2 Range: 1,1 On this limited interval, we have a one-to-one function.3

INVERSE SINE FUNCTIONHere’s the graph of restricted sine function:Restricted Sine functionf ( x) sin( x)Inverse Sine Function:sin 1 ( x)orarcsin( x) Domain: , 2 2 Domain: 1,1 Range: 1,1 Range: , (quadrants 1 and 4) 2 2 Example: 1 1 sin sin 1 6 2 2 64

1 Example: sin 1 ? 2 Question: What is the angle in , whose sine is 1/2? 2 2 Example: sin 1 1 ? Question: What is the angle in , whose sine is 1? 2 2 3 Example: sin 1 ?2 3 Question: What is the angle in , whose sine is?2 2 2 5

Important: When we covered the unit circle, we saw that there were two anglesthat had the same value for most of our angles. With inverse trig functions, thiswill not happen since we start with restricted functions that are one-to-one. We’llhave one quadrant in which the values are positive and one quadrant where thevalues are negative. The restricted graphs we looked at can help us know wherethese values lie. We’ll only state the values that lie in these intervals (same as theintervals for our graphs): 1 5 1Example: sin and sin ;626 2 5 1 However, sin 1 (unique answer!) sinceis Not in the range of6 2 6inverse since function.6

INVERSE COSINE FUNCTIONLet’s do the same thing with f ( x) cos( x) .Here’s the graph of f ( x) cos( x) .Domain: , Range: 1,1 It’s not one-to-one. If we limit the function to the interval 0, , however, thefunction IS one-to-one.7

Here’s the graph of the restricted cosine function.Restricted Cosine functionDomain: 0, Range: 1,1 8

INVERSE COSINE FUNCTIONNow, let’s work on defining the inverse of cosine function.Here’s the graph of restricted cosine function:Restricted Cosine functionf ( x) cos( x)Inverse Cosine Function:cos 1 ( x) or arccos( x)Domain: 0, Domain: 1,1 Range: 1,1 Range: 0, (quadrants 1 and 2) 1 1 cos 1 Example: cos 3 2 2 39

1 Example: cos 1 ? 2 Question: What is the angle in 0, whose cosine is1?2 2 Example: cos 1 ? 2 Question: What is the angle in 0, whose cosine is2?2Example: cos 1 1 ?Question: What is the angle in 0, whose cosine is 1?10

POPPER for Section 5.4:Question#1: What is the RANGE of g ( x) cos 1 ( x) ? A) , 2 2 B) , 2 2 C) 0, D) 0, E) 1,1 F) None of these11

INVERSE TANGENT FUNCTIONHere’s the graph of f ( x) tan( x) . Is it one-to-one? If we restrict the function to the interval , , then the restricted function IS 2 2 one-to-one.12

Restricted Tangent function Domain: , 2 2 Range: , 13

Inverse Tangent FunctionHere’s the graph of restricted tangent function:Restricted Tangent functionf ( x) sin( x) Domain: , 2 2 Range: , Inverse Tangent Function:tan 1 ( x) , 2 2 orarctan( x)Domain: , Range: , (quadrants 1 and 4) 2 2 Example: tan 1 tan 1 1 4 4 14

Example 1: Compute each of the following:a) cos 1 (0) b) tan 1 3 1 c) sin 1 2 2 d) sin 1 2 2 e) arccos 2 16

Example 2: Compute 1 arcsin arccos 0 arctan 1 2 17

POPPER for Section 5.4Question#2: Find the following sum: 2 arcsin arccos 0 2 a)b)c)d)e)f) 43 45 43 40None of these18

Note: If you need to compute inverse secant or inverse cosecant functions:Question: sec 1 (2) ?First, call it an angle:sec 1 (2) Then, convert:sec( ) 2Now, express this in terms of cosine: cos( ) 12And answer according to “inverse cosine function”: Final answer: sec 1 (2) 3 3Question: csc 1 (1) ?First, call it an angle:csc 1 (1) Then, convert:csc( ) 1Now, express this in terms of sine: sin( ) 1And answer according to “inverse sine function”: Final answer: csc 1 (1) 2 219

NOTE: Domains of inverse trig functions:f ( x) sin 1 ( x) ;[-1,1]f ( x) cos 1 ( x) ;[-1,1]f ( x) tan 1 ( x) ;( , )f ( x) cot 1 ( x) ;( , )f ( x) sec 1 ( x) ;f ( x) csc 1 ( x) ;( ,1] [1, )( ,1] [1, )For example; sin 1 (2) or cos 1 2 are not defined.20

Composition of a trig function with its inverse: 7 Example 3: Find the exact value: sin 1 sin 6 . 4 Example 4: Find the exact value: cos 1 cos . 3 3 Example 5: Find the exact value: tan 1 tan 4 . 21

Note: If a trigonometric function and its inverse are composed, then we have ashortcut. However, we need to be careful about giving an answer that is in therange of the inverse trig function.Examples: sin 1 sin 8 8but 7 sin 1 sin 8 7 8 cos 1 cos 8 8but 9 cos 1 cos 8 9 8 tan 1 tan 8 8but 7 7 tan 1 tan 8 8 22

If the inverse trig function is the inner function, then our job is easier:Examples: 1 1sin sin 1 5 5 2 2 cos cos 1 7 7 tan[tan 1 5 ] 5 .23

POPPER for Section 5.4 4 Question#3: Evaluate: sin sin 1 5 a)35b) c)254545e) UNDEFINEDf) None of thesed) 24

Let’s work with composition of different trig and inverse trig functions: 5 Example 6: Find the exact value: cos sin 1 . 13 25

2 Example 7: Find the exact value: tan cot 1 5 26

4 Example 8: Find the exact value: tan cos 1 . 5 27

1 Example 9: Find the exact value: sin arccos 4 28

Example 10: Find the exact value: tan sec 1 2 29

Example 11: Find the exact value: sin cos 1 4 30

x Example 12: Let y arctan where x 0 . Express 4 cos y in terms of x .31

POPPER for Section 5.4 3 Question#4: Evaluate: tan sin 1 5 a)b)c)d)e)f)35344543UNDEFINEDNone of these32

Graphs of Inverse Trigonometric FunctionsWe note the inverse sine function as f ( x) sin 1 ( x) or f ( x) arcsin( x).Domain: [-1, 1] Range: , . 22 Key points: 1, , 0,0 , 1, 2 2 Here is the graph of f ( x) sin 1 ( x) :33

Inverse Cosine FunctionWe note the inverse cosine function as f ( x) cos 1 ( x) or f ( x) arccos( x).Domain: 1,1 Range: 0, Key points: ( 1, ), 0, , 1,0 2 Here is the graph of f ( x) cos 1 ( x) :34

Inverse Tangent Function:We note the function as f ( x) tan 1 ( x) or f ( x) arctan( x).Domain: , Range: , 2 2 Key points: 1, ,(0,0), 1, 4 4 Here is the graph of the inverse tangent function:Important:Inverse tangent function has two horizontal asymptotes: y 2and y 2.You can use graphing techniques learned in earlier lessons to graphtransformations of the basic inverse trig functions.35

Example 1: Which of the following points is on the graph of f ( x) arctan( x 1) ? A) ,0 4 B) 0, 4 C) 0, 4 D) 2, 4 36

Example 2: Which of the following can be the function whose graph is givenbelow?A)B)C)D)E)f ( x) cos 1 ( x 1)f ( x) sin 1 ( x 1)f ( x) cos 1 ( x 1)f ( x) sin 1 ( x 1)f ( x) tan 1 ( x 1)37

Example 3: Which of the following can be the function whose graph is givenbelow?A)B)C)D)E)f ( x) cos 1 ( x 2)f ( x) sin 1 ( x 2)f ( x) cos 1 ( x 2)f ( x) sin 1 ( x 2)f ( x) tan 1 ( x 2)38

POPPER for Section 5.4Question#5: Which of the following points is ON the graph off ( x) arcsin( x 2) ? A) 1, 4 B) 1,0 C) 2, D) 2,0 E) 1, F) None of these39

Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL – Facts about inverse functions: A function f ()x is one-to-one if no two different inputs produce the same output (or: passes the horizontal line test) Example: f ()xx 2 is NOT one-to-one. gx x() 3 is one-to-one.