Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
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1 Section Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs produce the same output or: passes the horizontal line test) Eample: f ) 2 is NOT one-to-one. g ) 3 is one-to-one. A function f ) is invertible if it is one-to-one. The inverse function is notated as: f 1 ). f : A B f 1 : B A Domain: A Range: B Domain: B Range: A Important: f a) b if and only if f 1 b) a. 1
2 INVERSE SINE FUNCTION Here s the graph of f ) sin ). Domain:, Range: 1,1 Is it one to one? If the function is not one-to-one, we run into problems when we consider the inverse of the function. What we want to do with the sine function is to restrict the values for sine. When we make a careful restriction, we can get something that IS one-to-one. 2
3 If we limit the function to the interval, 2 2, the graph will look like this: Restricted Sine function Domain:, 2 2 Range: 1,1 On this limited interval, we have a one-to-one function. 3
4 INVERSE SINE FUNCTION Here s the graph of restricted sine function: Restricted Sine function Inverse Sine Function: f ) sin ) 1 sin ) or arcsin ) Domain:, 2 2 Domain: 1,1 Range: 1,1 Range:, 2 2 quadrants 1 and 4) Eample: sin sin
5 Eample: sin? 2 Question: What is the angle in, whose sine is 1/2? Eample: 1 sin 1? Question: What is the angle in, 2 2 whose sine is 1? 1 3 Eample: sin? 2 Question: What is the angle in, 2 2 whose sine is 3 2? 5
6 Important: When we covered the unit circle, we saw that there were two angles that had the same value for most of our angles. With inverse trig functions, this will not happen since we start with restricted functions that are one-to-one. We ll have one quadrant in which the values are positive and one quadrant where the values are negative. The restricted graphs we looked at can help us know where these values lie. We ll only state the values that lie in these intervals same as the intervals for our graphs): Eample: sin and sin ; However, sin unique answer!) since inverse since function. is Not in the range of 6
7 INVERSE COSINE FUNCTION Let s do the same thing with f ) cos ). Here s the graph of f ) cos ). Domain:, Range: 1,1 It s not one-to-one. If we limit the function to the interval 0,, however, the function IS one-to-one. 7
8 Here s the graph of the restricted cosine function. Restricted Cosine function Domain: 0, Range: 1,1 8
9 INVERSE COSINE FUNCTION Now, let s work on defining the inverse of cosine function. Here s the graph of restricted cosine function: Restricted Cosine function Inverse Cosine Function: 1 f ) cos ) cos ) or arccos ) Domain: Range: 0, Domain: 1,1 1,1 Range: 0, quadrants 1 and 2) Eample: cos cos
10 Eample: cos? Question: What is the angle in 0, whose cosine is 1 2? 1 2 Eample: cos? 2 Question: What is the angle in 0, whose cosine is 2 2? Eample: 1 cos 1? Question: What is the angle in 0, whose cosine is 1? 10
11 POPPER for Section 5.4: Question#1: What is the RANGE of g 1 ) cos )? A), 2 2 B), 2 2 0, C) D) 0, E) 1,1 F) None of these 11
12 INVERSE TANGENT FUNCTION Here s the graph of f ) tan ). Is it one-to-one? If we restrict the function to the interval,, then the restricted function IS 2 2 one-to-one. 12
13 Restricted Tangent function Domain:, 2 2 Range:, 13
14 Inverse Tangent Function Here s the graph of restricted tangent function: Restricted Tangent function Inverse Tangent Function: f ) sin ) 1 tan ) or arctan ) Domain:, 2 2 Range:,, 2 2 Domain:, Range:, 2 2 quadrants 1 and 4) Eample: tan 1 tan 1 14
15
16 Eample 1: Compute each of the following: a) 1 cos 0) 1 b) tan 3 c) sin d) sin e) arccos
17 Eample 2: Compute 1 arcsin arccos 0 arctan
18 POPPER for Section 5.4 Question#2: Find the following sum: a) 4 b) 3 4 c) d) 4 e) 0 f) None of these 2 arcsin arccos
19 Note: If you need to compute inverse secant or inverse cosecant functions: Question: 1 sec 2)? First, call it an angle: 1 sec 2) Then, convert: sec ) 2 Now, epress this in terms of cosine: 1 cos ) 2 And answer according to inverse cosine function : 3 Final answer: 1 sec 2) 3 Question: 1 csc 1)? First, call it an angle: 1 csc 1) Then, convert: csc ) 1 Now, epress this in terms of sine: sin ) 1 And answer according to inverse sine function : 2 Final answer: 1 csc 1) 2 19
20 NOTE: Domains of inverse trig functions: f ) sin ) ; [-1,1] f ) cos ) ; [-1,1] 1 f ) tan ) ;, ) 1 f ) cot );, ) 1 f ) sec ) ;,1] [1, ) 1 f ) csc ) ;,1] [1, ) For eample; sin 1 2) or cos 1 2 are not defined. 20
21 Composition of a trig function with its inverse: Eample 3: Find the eact value: 1 7 sin sin. 6 Eample 4: Find the eact value: 1 4 cos cos. 3 Eample 5: Find the eact value: 1 3 tan tan. 4 21
22 22 Note: If a trigonometric function and its inverse are composed, then we have a shortcut. However, we need to be careful about giving an answer that is in the range of the inverse trig function. Eamples: 8 8 sin sin 1 but sin sin cos cos 1 but cos cos tan tan 1 but tan tan 1
23 If the inverse trig function is the inner function, then our job is easier: Eamples: sinsin coscos tan[tan 1 5 ] 5. 23
24 POPPER for Section 5.4 Question#3: Evaluate: sinsin a) 3 5 b) 2 5 c) d) 5 e) UNDEFINED f) None of these 24
25 Let s work with composition of different trig and inverse trig functions: 1 5 Eample 6: Find the eact value: cossin
26 Eample 7: Find the eact value: tancot
27 Eample 8: Find the eact value: 1 4 tan cos. 5 27
28 Eample 9: Find the eact value: 1 sinarccos 4 28
29 Eample 10: Find the eact value: 1 tan sec 2 29
30 1 Eample 11: Find the eact value: sin cos 4 30
31 Eample 12: Let y arctan 4 where 0. Epress cos y in terms of. 31
32 POPPER for Section 5.4 Question#4: Evaluate: tansin a) 3 5 b) 3 4 c) 4 5 d) 4 3 e) UNDEFINED f) None of these 32
33 Graphs of Inverse Trigonometric Functions 1 We note the inverse sine function as f ) sin ) or f ) arcsin ). Domain: [-1, 1] Range:, 2 2. Key points: 1,, 0,0, 1, 2 2 Here is the graph of f ) sin 1 ) : 33
34 Inverse Cosine Function 1 We note the inverse cosine function as f ) cos ) or f ) arccos ). Domain: 1,1 Range: 0, Key points: 1, ), 0,, 1,0 2 Here is the graph of f ) cos 1 ) : 34
35 Inverse Tangent Function: 1 We note the function as f ) tan ) or f ) arctan ). 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 and 2 y. 2 You can use graphing techniques learned in earlier lessons to graph transformations of the basic inverse trig functions. 35
36 Eample 1: Which of the following points is on the graph of f ) arctan 1)? A),0 4 B) 0, 4 C) 0, 4 D) 2, 4 36
37 37 Eample 2: Which of the following can be the function whose graph is given below? A) ) 1 cos ) 1 f B) ) 1 sin ) 1 f C) ) 1 cos ) 1 f D) ) 1 sin ) 1 f E) ) 1 tan ) 1 f
38 38 Eample 3: Which of the following can be the function whose graph is given below? A) ) 2 cos ) 1 f B) ) 2 sin ) 1 f C) ) 2 cos ) 1 f D) ) 2 sin ) 1 f E) ) 2 tan ) 1 f
39 POPPER for Section 5.4 Question#5: Which of the following points is ON the graph of f ) arcsin 2)? A) 1, 4 1, 0 B) C) 2, D) 2,0 E) 1, F) None of these 39
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