6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives

Size: px

Start display at page:

Download "6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives"

Trevor Banks
5 years ago
Views:

1 Objectives 1. Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function. 2. Find an Approximate Value of an Inverse Sine Function. 3. Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions. 4. Find the Inverse Function of a Trigonometric Function. 5. Solve Equations Involving Inverse Trigonometric Functions. 7 May Kidoguchi, Kenneth

2 Review of Properties of Inverse Functions (cf. 4.2) If f(x) is a function that is one-to-one on a specified interval, then: 1. f -1 (f(x)) = x for every x in the domain of f and f (f -1 (x)) = x for every x in the domain of f Domain of f = range of f -1 and range of f = domain of f The graph of f and the graph of f -1 are symmetric with respect to the identity line y = x. 4. If a function y = f(x) has an inverse function, the implicit equation of the inverse function is x = f(y). If we solve this equation for y, we obtain the explicit equation y = f -1 (x). 7 May Kidoguchi, Kenneth

3 An Algebraic Recipe for Finding Inverse Functions 0. Verify that the given expression is a function. 1. Verify that the function is one-to-one. If the given expression is not a one-to-one function, but can be made one-to-one by specifying constraints on its domain, do so. 2. y = f(x) : Introduce the dummy variable y. 3. x = f(y) : Interchange x and y in step y = g(x): Solve equation in step 3 for y (i.e., make y the subject of the equation). 5. Define the inverse function, g(x) = f 1 (x). 6. Verify that f(g(x)) = g(f(x)) = x. Graphically, this means that the graph of the inverse function is a reflection of the original function about the identity line. 7 May Kidoguchi Kenneth

4 An Algebraic Recipe for Finding Inverse Functions - Example Consider: f(x) = x 2 0. Verify that the given expression is a function. 1. Verify that the function is one-toone. If the given expression is not a one-to-one function, but can be made one-to-one by specifying constraints on its domain, do so. 7 May Kidoguchi Kenneth

5 An Algebraic Recipe for Finding Inverse Functions - Example 2. y = f(x) : Introduce the dummy variable y. 3. x = f(y) : Interchange x and y in step y = g(x): Solve equation in step 3 for y (i.e., make y the subject of the equation). 5. Define the inverse function, g(x) = f 1 (x). 6. Verify that f( g(x) ) = g( f(x) ) = x. Graphically, this means that the graph of the inverse function is a reflection of the original function about the identity line. 7 May Kidoguchi, Kenneth

6 7 May Kidoguchi, Kenneth

7 Sine Function Default One-to-One Interval The function sin(x) is not one-to-one on the interval: < x <. There are an infinite number of choices that would make sin(x) a one-toone function. The default sin(t) one-to-one interval is: /2 x /2. y x 7 May Kidoguchi Kenneth

8 Cosine Function Default One-to-One Interval The function cos(x) is not one-to-one on the interval: < x <. There are an infinite number of choices that would make cos(x) a oneto-one function. The default cos(x) one-to-one interval is: 0 x x 7 May Kidoguchi Kenneth

9 Tangent Function Default One-to-One Interval The function tan(x) is not one-to-one on the interval: < x <. There are an infinite number of choices that would make tan(x) a oneto-one function. The default tan(x) one-to-one interval is: /2 x /2. y t 7 May Kidoguchi Kenneth

10 The Inverse Sine Function Textbook notation: 1 ( ) sin ( ) y = sin x x= y 1 x 1 y 2 2 Alternate Notation: ( ) sin ( ) y = arcsin x x= y 1 x 1 y May Kidoguchi, Kenneth

11 The Inverse Cosine Function Textbook notation: 1 ( ) cos( ) y = cos x x= y 1 x 1 0 y Alternate Notation: ( ) cos( ) y = arccos x x= y 1 x 1 0 y 7 May Kidoguchi, Kenneth

12 The Inverse Tangent Function Textbook notation: 1 ( ) tan ( ) y = tan x x= y x y 2 2 Alternate Notation: ( ) tan ( ) y = arctan x x= y x y May Kidoguchi, Kenneth

13 Properties sin 1 ( x ) = arcsin( x) The angle whose sine is x sin(arcsin( x )) = x For unspecified domain, assume: /2 x /2 cos 1 ( x ) = arccos( x) The angle whose cosine is x cos(arccos( x )) = x For unspecified domain, assume: 0 x tan 1 ( x ) = arctan( x) The angle whose tangent is x For unspecified domain, assume: tan(arctan( x )) = /2 x /2 x 7 May Kidoguchi Kenneth

14 (a) 6.1 The Inverse Sine, Cosine, and Tangent Functions 1. Find the Exact Value of an Inverse Sine Function Find the exact value of: (b) θ= sin θ= sin θ sin(θ) N.B.: the default one-to-one interval for sin(θ) is [/2, /2]. 7 May Kidoguchi, Kenneth

15 2. Find the Approximate Value of an Inverse Sine Function Find an approximate value of: (a) sin (b) sin N.B.: the default one-to-one interval for sin(θ) is [/2, /2]. 7 May Kidoguchi, Kenneth

16 1. Find the Exact Value of an Inverse Cosine Function Find the exact value of: (a) cos 1 2 = 2 (b) cos 1 1 = 2 N.B.: The default one-to-one interval for cos(θ) is [0, ]. 7 May Kidoguchi, Kenneth

1. Find the Exact Value of an Inverse Tangent Function Find the exact value of: 3 3 1 (a) tan = ( ) 1

17 1. Find the Exact Value of an Inverse Tangent Function Find the exact value of: (a) tan = ( ) 1 (b) tan 1 = N.B.: the default one-to-one interval for tan(θ) is [/2, /2]. 7 May Kidoguchi, Kenneth

18 3. Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions Find the exact value of the given composite function: 12-1 (a) sin sin = 1 (b) sin sin 5 3 = ( -1 ( )) -1 ( ) (c) sin sin 0.4 = ( ) (d) sin sin 1.2 = 1 2 (e) cos cos = 3 ( ) ( -1 ) (f) cos cos 0.6 = ( ) ( -1 ) (g) cos cos 1.8 = 7 May Kidoguchi, Kenneth

19 Finding the Inverse of a Trigonometric Function Find the inverse function f -1 of f(x) = -3 cos(x) + 2 and state the domain of f and f May Kidoguchi, Kenneth

20 Finding the Inverse of a Trigonometric Function Find the inverse function f -1 of f(x) = -3 cos(x) + 2 and state the domain of f and f -1. ( x) ( x) y = 3cos + 2 3cos = 2 y 2 y cos( x) = 3 2 x cos( y) = 3 2 x arccos ( cos( y) ) = arccos 3 2 x y = arccos x f ( x) = arccos 3 7 May Kidoguchi, Kenneth

Finding the Inverse of a Trigonometric Function f 1 ( x) f( x) = 3cos + 2 2 x ( x) = arccos 3 The domain

21 Finding the Inverse of a Trigonometric Function f 1 ( x) f( x) = 3cos x ( x) = arccos 3 The domain of f -1 2 x x 3 5 x 1 1 x 5 The domain of f 1 1 f ( 1) x f (5) 0 x 7 May Kidoguchi, Kenneth

22 5. Solve Equations Involving Inverse Trigonometric Function Solve the equation: tan -1 (x) 5 tan -1 (x) /3 = 2/3. 7 May Kidoguchi, Kenneth

23 5. Solve Equations Involving Inverse Trigonometric Function Solve the equation: tan -1 (x) 5 tan -1 (x) /3 = 2/3. 4arctan( x) = arctan( x) = 4 tan ( arctan( x) ) = tan x = May Kidoguchi, Kenneth

SET 1. (1) Solve for x: (a) e 2x = 5 3x

SET 1. (1) Solve for x: (a) e 2x = 5 3x () Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x