Inverse Trigonometric Functions

Inverse Trigonometric Functions answers the question: given the value of a trigonometric ratio, what is the angle? Since trigonometric functions repeat their values, they are not one-one on their natural domains, so their inverses exist only when we restrict each function to a suitable interval. In this chapter you will learn how restricting sine to [-pi/2, pi/2], cosine to [0, pi] and tangent to (-pi/2, pi/2) makes them one-one and onto, giving the inverse functions sin^-1 x, cos^-1 x, tan^-1 x, cosec^-1 x, sec^-1 x and cot^-1 x. You will learn the domain and range of each inverse function, the meaning of the principal value branch, and how to find principal values such as sin^-1(-1/2) = -pi/6 and cos^-1(-1/2) = 2pi/3. You will draw the graphs of the inverse functions as mirror images of the restricted trigonometric graphs in the line y = x, and evaluate expressions such as sin^-1(sin(3pi/5)), where you must bring the angle back into the principal branch.

Detailed Worksheet: Inverse Trigonometric Functions Section A - Definitions (10 marks) 1. Why does the sine function not have an inverse on R? How is its domain restricted to define sin^-1 x? (2 marks) 2. Define the principal value branch of cos^-1 x. State its domain and range. (2 marks) 3. Write the domain and range of tan^-1 x and cot^-1 x. (2 marks) 4. Is sin^-1 x the same as (sin x)^-1? Explain. (2 marks) 5. State the domain and principal value range of sec^-1 x and cosec^-1 x. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the principal values of (i) sin^-1(-1/2), (ii) cos^-1(-1/2) and (iii) tan^-1(-sqrt(3)). (3 marks) 7. Find the value of tan^-1(1) + cos^-1(-1/2) + sin^-1(-1/2). (3 marks) 8. Evaluate (i) sin^-1(sin(3pi/5)), (ii) cos^-1(cos(13pi/6)) and (iii) tan^-1(tan(7pi/6)). (3 marks) 9. Find the domain of (i) sin^-1(2x) and (ii) cos^-1(x^2 - 4). (3 marks) 10. Find the value of (i) sin(pi/3 - sin^-1(-1/2)) and (ii) cos(sin^-1(3/5)), using a right triangle for (ii). (3 marks) Section C - Diagrams (10 marks) 11. Draw the graph of y = sin x on [-pi/2, pi/2] and of y = sin^-1 x on [-1, 1] on the same axes. Show that one is the mirror image of the other in the line y = x. (4 marks) 12. Draw the graph of y = cos^-1 x for -1 <= x <= 1 and mark its range. (3 marks) 13. Draw the graph of y = tan^-1 x and mark the horizontal asymptotes y = pi/2 and y = -pi/2. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A student writes sin^-1(sin(5pi/6)) = 5pi/6 and cos^-1(cos(7pi/6)) = 7pi/6. Identify the errors, find the correct values, and explain the rule for bringing an angle into the principal branch. (5 marks) 15. Find the values of (i) tan^-1(sqrt(3)) - sec^-1(-2), (ii) cos^-1(1/2) + 2 sin^-1(1/2), (iii) sin(tan^-1(3/4)) and (iv) tan^-1(tan(3pi/4)), giving reasons. (5 marks) 16. Explain why the branch [-pi/2, pi/2] is chosen as the principal branch of sin^-1 x but [0, pi] is chosen for cos^-1 x. Could [pi/2, 3pi/2] also serve as a branch of sin^-1 x? Discuss with reference to one-one and onto functions and sketch the alternative branch. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Give all angles in radians within the principal value branch. Draw neat graphs with labelled axes.