Section 5.4 - Inverse Trigonometric Functions The Inverse Sine Function Consider the graph of the sine function f ( x) sin( x). The function is a periodic function. That means that the functions repeats its values in regular intervals, which we call the period. 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. 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 sine function. 1
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of sine: f ( x) sin ( x) or f ( x) arcsin( x). Restricted Sine Function (Blue) Inverse Sine Function (Red) Domain:, 2 2 Domain: 1,1 Range: 1,1 Range:, 2 2 (quadrants 1 and 4) To find the value of sin 1 ( x), find the number in the interval, 2 2whose sine is x. Example: 1 1 1 sin sin 6 2 2 6 2
The Inverse Cosine Function Let s do the same thing with the cosine function f ( x) cos( x), which is not one-to-one. Here s the graph of f ( x) cos( x). If we limit the function to the interval 0,, however, the function IS one-to-one. Here s the graph of the restricted cosine function. Restricted Cosine Function Domain: 0, Range: 1,1 On this limited interval, we have a one-to-one cosine function. 3
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of cosine: f ( x) cos ( x) or f ( x) arccos( x). Restricted Cosine Function (Blue) Inverse Cosine Function (Red) Domain: Range: 0, Domain: 1,1 1,1 Range: 0, (quadrants 1 and 2) To find the value of cos 1 ( x), find the number in the interval 0, whose cosine is x. Example: 2 1 2 cos cos 4 2 2 4 4
The Inverse Tangent Function Here s the graph of tangent function f ( x) tan( x), which is not one-to-one: If we restrict the function to the interval,, then the function IS one-to-one. 2 2 Restricted Tangent Function Domain:, 2 2 Range:, On this limited interval, we have a one-to-one tangent function. 5
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of tangent: f ( x) tan ( x) or f ( x) arctan( x). Restricted Tangent Function (Blue) Domain:, 2 2 Inverse Tangent Function (Red) Domain:, Range:, Range:, 2 2 (quadrants 1 and 4) To find the value of tan 1 ( x), find the number in the interval, whose tangent is x. 2 2 Example: 4 4 1 tan 1 tan 1 6
The Inverse Cotangent Function Here s the graph of cotangent function f ( x) cot( x), which is not one-to-one: If we restrict the function to the interval ( 0, ), then the function IS one-to-one. Restricted Cotangent Function Domain: ( 0, ) Range:, On this limited interval, we have a one-to-one cotangent function. 7
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of cotangent: f ( x) cot ( x) or f ( x) arc cot( x). Restricted Cotangent Function (Blue) Inverse Cotangent Function (Red) Domain: 0, ) ( Domain:, Range:, Range: ( 0, ) (quadrants 1 and 2) To find the value of cot 1 ( x), find the number in the interval ( 0, ) whose cotangent is x. 6 6 Example: cot 3 1 cot 3
The Inverse Secant Function Here s the graph of secant function f ( x) sec( x), which is not one-to-one: If we restrict the function to the interval 0,, 2 2, then the function IS one-to-one. Restricted Secant Function Domain: 0,, 2 2 Range:, 1 1, On this limited interval, we have a one-to-one secant function. 9
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of secant: f ( x) sec ( x) or f ( x) arc sec( x). Restricted Secant Function (Blue) Inverse Secant Function (Red) Domain:,, 2 2 0 Domain:, 1 1, Range: 1 1,, Range: 0,, 2 2 (quadrants 1 and 2) To find the value of sec 1 ( x) whose secant is x., find the number in the interval 0,, 2 2 3 3 Example: sec 1 2 sec 2 10
The Inverse Cosecant Function Here s the graph of cosecant function f ( x) csc( x), which is not one-to-one: If we restrict the function to interval,0 0, 2 2, then the function IS one-to-one. Restricted Tangent Function Domain:,0 0, 2 2 Range:, 1 1, On this limited interval, we have a one-to-one cosecant function. 11
Being an one-to-one function implies that the inverse of it does exist. 1 Notation for the inverse of cosecant: f ( x) csc ( x) or f ( x) arc csc( x). Restricted Cosecant Function (Blue) Domain:,0 0, 2 2 Inverse Cosecant Function (Red) Domain:, 1 1, Range: 1 1,, Range:,0 0, 2 2 (quadrants 1 and 4) To find the value of csc 1 ( x) whose cosecant is x., find the number in the interval,0 0, 2 2 6 6 Example: csc 1 2 csc 2 12
Note: We always give inverse trigonometric angles in radians. Example 1: Compute each of the following: a) sin 2 1 1 1 b) tan 3 c) arccos(0) d) sin 1 2 2. e) sin 2 1 1 f) 3 cos 1. 2 g) arctan( 1). h) sec 1 (2). i) csc 1 (0) 13
NOTE: Domains of inverse trig functions: 1 x f ( x) sin ( ) ; [-1,1] 1 x f ( x) cos ( ) ; [-1,1] 1 x f ( x) tan ( ) ; (, ) f ( x) cot 1 ( x) ; (, ) 1 x f ( x) sec ( ) ; (, 1] [1, ) 1 x f ( x) csc ( ) ; (, 1] [1, ) For example; sin 1 (2) or cos 1 2 are not defined. 14
Composition of Trigonometric Functions with their Inverses and viceversa Recall that if f and g are inverse functions then the following hold: f ( g( x)) x for each x in the domain of g. g( f ( x)) x for each x in the domain of f. When a trigonometric function and its inverse are composed, we need to be careful about giving an answer that is in the range of the inverse trig function. cos x x cos 1 if x [ 0, ] sin x x if x, 2 2 tan 1 tan x x if x, 2 2 sin 1 Examples: sin 1 sin but 7 7 sin 1 sin cos 1 cos but 9 9 cos 1 cos tan 1 tan but 7 7 tan 1 tan If the inverse trigonometric function is the inner function, then our job is easier. x] x] x] cos[cos 1 x for any number x such that 1 x 1. sin[sin 1 x for any number x such that 1 x 1. tan[tan 1 x for any number x. Examples: sin sin 1 1 5 1 5 coscos 1 2 2 7 7 1 1 tan[tan 1 ] 4 4 tan[tan 1 5 ] 5. 15
Example 2: Find the exact value of the following: a) 1 7 sin sin. 6 b) 1 4 cos cos. 3 c) 1 3 tan tan. 4 d) 5 arcsin sin 3 e) tan tan 3 1 2 Example 3: Find the exact value of the following: a) 1 1 cos cos. 6 1 5 b) cossin. 13 16
c) 1 2 tan cos. 5 d) tan cot 5 1 2 e) sin cos 4 1 1 f) tan sin 5 1 4 g) sec 1 tan 2 Example 4: Simplify 1 cos arctan x 4 where x 0. 17
Working with Graphs of Inverse Trigonometric Functions Here are the graphs of the trigonometric functions: We can use graphing techniques learned in previous lectures to graph transformations of the basic inverse trig functions. 1
Example 5: Which of the following points is on the graph of f ( x) arctan( x 1)? A),0 4 B) 0, C) 0, D) 2, 4 4 4 Example 6: Which of the following can be the function whose graph is given below? 1 A) f ( x) cos ( x 1) 1 B) f ( x) sin ( x 1) 1 C) f ( x) cos ( x 1) 1 D) f ( x) sin ( x 1) 1 E) f ( x) tan ( x 1) Example 7: Which of the following can be the function whose graph is given below? 1 A) f ( x) cos ( x 2) 1 B) f ( x) sin ( x 2) 1 C) f ( x) cos ( x 2) 1 D) f ( x) sin ( x 2) 1 E) f ( x) tan ( x 2) 19
Modeling Using Sinusoidal Functions Sine and cosine functions model many real-world situations. Physical phenomenon such as tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily modeled by sine and cosine functions (collectively, they are called sinusoidal functions ). Here are some other situations that can be modeled by a sinusoidal function: Suppose you are on a Ferris wheel at a carnival. Your height (as you are sitting in your seat) varies sinusoidally. Suppose you are pushing your child as s/he sits in a swing. Your child s height varies sinusoidally. The motion of a swinging pendulum varies sinusoidally. Stock prices sometimes vary sinusoidally. We ll work a couple of examples involving sinusoidal variation. Recall: Given the functions f ( x) Asin( Bx C) D or f ( x) Acos( Bx C) D ; 2 The period is: B The amplitude is: A The vertical shift is: D (up if positive) The horizontal shift is B C (to the right if positive) Example : Determine the equation of the sine function which has amplitude is 5, the phase shift is 4 to the left, the vertical shift is 3 down, and the period is 2. 20
Example 9: The number of hours of daylight in Boston is given by 2 f ( x) 3sin ( x 79) 12 where x is the number of days after January 1. What is the: 365 a. amplitude? b. period? c. maximum value of f(x)? Example 10: The function P( t) 120 40 sin(2 t) models the blood pressure (in millimeters of mercury) for a person who has a blood pressure of 160/90 (which is high); t represents seconds. What is the period of this function? What is the amplitude? 21
Example 11: Determine the function of the form f ( x) Asin( Bx) given the following graph: Example 12: Determine the function of the form f ( x) Asin( Bx C) D given the graph: 22
Example 13: Assume that you are aboard a research submarine doing submerged training exercises in the Pacific Ocean. At time t = 0 you start purposing (alternately deeper and then shallower). At time t = 4 min you are at your deepest, y = 1000 m. At time t = 9 min you next reach your shallowest, y = 200 m. Assume that y varies sinusoidally with time. Find an equation expressing y as a function of t. 5 f ( t) 400cos t 4 5 f ( t) 200cos t 4 5 f ( t) 600cos t 4 5 f ( t) 400cos t 9 15 A) f ( t) 600cos t 4 200 B) 600 C) 400 D) 400 E) A) 600 (Extra) Example: A signal buoy in the Gulf of Mexico bobs up and down with the height h of its transmitter (in feet) above sea level modeled by h ( t) Asin( Bt) 5. During a small squall its height varies from 1 ft to 9 ft and there are 4 seconds from one 9-ft height to the next. What are the values of the constants A and B? 23