Unit 3 Eploring Inverse trig. functions Standards: F.BF. Find inverse functions. F.BF.d (+) Produce an invertible function from a non invertible function by restricting the domain. F.TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it s inverse to be constructed. F.TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contets; evaluate the solutions using technology, and interpret them in terms of the contet. Essential Question(s): Why does the calculator only give one answer for an inverse trig function? Aren't there infinite answers? How do inverse trigonometric functions help us solve equations? Notes 1. If the point (, y) is on the function, the point (y, ) is on the inverse.. Two functions are inverses of one another if and only if f(f -1 ()) = and f -1 (f()) =. 3. The inverse of a function will reflect over the line y =.. In order for a graph to be a function, it must pass the vertical line test (VLT). 5. In order for a function to have an inverse function, it must pass the horizontal line test (HLT). Eample #1: f() =. Sketch the graph of f() and determine if it passes VLT and HLT. Is the inverse of f() a function? Eample #: Can we restrict f() = such that it passes both VLT and HLT. Sketch the new graph and write how the function would be restricted. Eample #3: Graph f(θ) = sin θ and the line y = 1. a. How many times do these functions intersect between - and? b. How is this graph related to finding the solution to 1 = sin θ? c. If the domain is not limited, how many solutions eist to the equation 1 = sin θ? d. Would this be true for the other trigonometric functions? Eplain. Eample #: f() = sin()? a. Complete the table for f() = sin(). Then sketch the graph of f() = sin and its reflection across the line y =. sin() 3
b. What does your graph tell you about the relationship between the graph of y = sin() and y = sin -1 ()? c. Is y = sin -1 () a function? d. What is the domain and range of y = sin()? e. Can we restrict the range of y = sin -1 () so that it would be a function? What would it need to be? f. Graph the function y = sin -1 () with the restricted range to the right. Eample #5: f() = cos() a. Complete the table, then graph the function f() = cos and its reflection over the line y =. cos() 3 b. What does the graph of above tell you about the relationship between the graph of y = cos() and y = cos -1 ()? c. Is y = cos -1 () a function? d. What is the domain and range of y = cos()? e. Can we restrict the range of y = cos -1 () so that it would be a function? What would it need to be? f. Graph the function y = cos -1 () with the restricted range to the right.
Eample #6: f() = tan() a. Complete the table, then graph the function f() = tan and its inverse. 3 5 tan() b. What does the graph of above tell you about the relationship between the graph of y = tan() and y = tan -1 ()? c. Is y = tan -1 () a function? d. What is the domain and range of y = tan()? e. Can we restrict the range of y = tan -1 () so that it would be a function? h. Graph the function y = tan -1 () with the restricted range. In Summary We use the names sin -1, cos -1, and tan -1 or ArcSin, ArcCos, and ArcTan to represent the inverse of these functions on the limited domains you eplored above. The values in the limited domains of sine, cosine and tangent are called principal values. (Similar to the principal values of the square root function.) Calculators give principal values when reporting sin -1, cos -1, and tan -1. Complete the chart below indicating the domain and range of the given functions. Function Domain Range f(θ) = sin 1 θ f(θ) = cos 1 θ f(θ) = tan 1 θ
Note: arcsin in the problem arcsin 1 means the angle whose sin is. Thus you are to find the sin of what is 1. We need to remember that arcsin is only defined in Quadrant I and IV, so we must find an answer in those quadrants only. Since the sin( 6 ) is 1 the answer is 6 The inverse functions do not have ranges that include all quadrants. Add a column to your chart that indicates the quadrants included in the range of the function. This will be important to remember when you are determining values of the inverse functions. 1. Use what you know about trigonometric functions and their inverses to evaluate the following epressions. Two eamples are included for you. (Unit circles can also be useful.) Eample 1: ArcCos ( 3 ) The answers will be an angle. Let θ = Arccos ( 3 ) Ask yourself, what angle has a cos value of 3. Cos θ = ( 3 ) Using the definition of Arccos. θ = ( 6 ) So, ArcCos ( 3 ) = ( 6 ) Why isn t (5 6 ) included? Eample : sin (cos -1 1 + tan -1 1) θ = cos -1 1 θ = tan -1 1 θ = θ = sin ( + ) sin( ) = So, sin (cos -1 1 + tan -1 1) = The answer will be a number, not an angle. Simplify parentheses first. Substitution Inverse Properties of Trigonometric Functions If -1 < < 1 and - y, then sin(arcsin ) = and arcsin(sin y) = y If -1 < < 1 and y, then cos(arccos ) = and arccos(cos y) = y If is a real number and - < y <, then tan(arctan ) = and arctan(tan y) = y Keep in mind, the properties such as arcsin(sin y) = y is not valid for values of y outside the interval [, ] Practice a. θ = cos 1 1 b. θ = Arcsin c. sin 1 d. cos (tan 1 3 sin 1 1 ) e. cos (tan 1 3 3 ) f. sin (sin 1 3 )
Work Period Without using a calculator (using only a unit circle), find the value of each epression in radians. Remember that the inverse trig functions have a restricted range. Make sure your answer falls in the range for that function. 1. sin 1. sin 1 1 3. sin 1 ( 1 ). sin 1 ( 5. sin 1 ( 1) 6. sin 1 ( 1 ) 7. sin 1 ( ) 8. cos 1 9. cos 1 1 1. cos 1 ( 1 ) 11. cos 1 ( 1. cos 1 ( 1) 13. cos 1 ( 1 ) 1. cos 1 ( ) 15. tan 1 16. tan 1 1 17. tan 1 ( 3 ) 18. tan 1 ( 3 3 19. tan 1 ( 1). tan 1 ( 3 ) 1. tan 1 ( 3 3 ) Homework Part 1. Complete the table Function Alternative Notation Domain Range y = arcsin y = arccos y = arctan 1. cos 1 ( 3 ). arcos ( ) 3. cos 1 (cos ( 3 )). arccos( 1 ) 5. cos 1 (cos ( 3 )) 6. sin 1 ( 3 7. sin(tan 1 3) 8. tan 1 (tan ( )) 9. sin 1 (cos ( )) 1. sin (cos 1 ( 1 )) 11. cos (tan 1 ( 3) sin 1 ( 3 )) 1. tan (cos 1 ( )) 13. cos 1 ( ) 1. sin 1 ( ) 15. sin 1 (sin ( 5 6 )) 16. cos 1 (sin ( 6 )) 17. sin 1 (tan ( 3 )) 18. arcsin() 19. cos 1 (cos ( 5 6 )). sin(cos 1 ( 3 )) Additional Practice Part 3 1. sin 1 ( 3 ). sin 1 ( 1 ) 3. tan 1. cos 1 1 5. cos 1 ( 1 ) 6. tan 1 1 7. tan 1 ( 1) 8. sin 1 ( 3 ) 9. sin 1 ( 1 ) 1. tan 1 ( 3) 11. cos 1 1. sin 1 (1) 13. cos(sin 1 ( 1 )) 1. sin(tan 1 ( 1 )) 15. sin 1 (cos ( )) 16. cos 1 (cos ( 7 )) 17. cos ( sin 1 ( 1 )) 18. sin(tan 1 ( 1 )) 19. arcsin (cos ( 3 )). arccos (tan ( )) 1. cos(tan 1 3). tan 1 (cos( ))