INVERSE TRIGONOMETRIC FUNCTIONS Notes

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1 Inverse Trigonometric s MODULE - VII INVERSE TRIGONOMETRIC FUNCTIONS In the previous lesson, you have studied the definition of a function and different kinds of functions. We have defined inverse function. Let us briefly recall : Let f be a one-one onto function from A to B. Let y be an arbitary element of B. Then, f being onto, an element A such that f y. Also, f being one-one, then must be unique. Thus for each y B, a unique element A such that f y. So we may define a function, Fig.. denoted by f as f : B A f y f y The above function f is called the inverse of f. A function is invertiable if and only if f is one-one onto. In this case the domain of f is the range of f and the range of f is the domain f. Let us take another eample. We define a function : f : Car Registration No. If we write, g : Registration No. Car, we see that the domain of f is range of g and the range of f is domain of g. So, we say g is an inverse function of f, i.e., g = f. In this lesson, we will learn more about inverse trigonometric function, its domain and range, and simplify epressions involving inverse trigonometric functions. OBJECTIVES After studying this lesson, you will be able to : define inverse trigonometric functions; state the condition for the inverse of trigonometric functions to eist; define the principal value of inverse trigonometric functions; find domain and range of inverse trigonometric functions; state the properties of inverse trigonometric functions; and simplify epressions involving inverse trigonometric functions. MATHEMATICS 5

2 MODULE - VII EXPECTED BACKGROUND KNOWLEDGE Inverse Trigonometric s Knowledge of function and their types, domain and range of a function Formulae for trigonometric functions of sum, difference, multiple and sub-multiples of angles.. IS INVERSE OF EVERY FUNCTION POSSIBLE? Take two ordered pairs of a function, y and, y If we invert them, we will get y, and y, This is not a function because the first member of the two ordered pairs is the same. Now let us take another function : sin,, sin, and 3 sin, 3 Writing the inverse, we have, sin,, sin and 3, sin 3 which is a function. Let us consider some eamples from daily life. f : Student Score in Mathematics Do you think f will eist? It may or may not be because the moment two students have the same score, f will cease to be a function. Because the first element in two or more ordered pairs will be the same. So we conclude that every function is not invertible. Eample. If f : R R defined by 3 f (). What will be f? Solution : In this case f is one-to-one and onto both. f is invertible. Let 3 y y y 3 3 So f, inverse function of f i.e., f y 3 y The functions that are one-to-one and onto will be invertible. Let us etend this to trigonometry : Take y = sin. Here domain is the set of all real numbers. Range is the set of all real numbers lying between and, including and i.e. y. 6 MATHEMATICS

3 Inverse Trigonometric s We know that there is a unique value of y for each given number. In inverse process we wish to know a number corresponding to a particular value of the sine. Suppose y sin 5 3 sin sin sin sin may have the values as,, Thus there are infinite number of values of. y = sin can be represented as, 6, 5,,... 6 The inverse relation will be,, 6 5,,... 6 MODULE - VII It is evident that it is not a function as first element of all the ordered pairs is, which contradicts the definition of a function. Consider y = sin, where R (domain) and y [, ] or y which is called range. This is many-to-one and onto function, therefore it is not invertible. Can y = sin be made invertible and how? Yes, if we restrict its domain in such a way that it becomes one-to-one and onto taking as (i) (ii) (iii), y [, ] or 3 5 y [, ] or 5 3 y [, ] etc. Now consider the inverse function y sin. We know the domain and range of the function. We interchange domain and range for the inverse of the function. Therefore, (i) (ii) y [, ] or 3 5 y [, ] or MATHEMATICS 7

4 MODULE - VII (iii) 5 3 y [, ] etc. Inverse Trigonometric s Here we take the least numerical value among all the values of the real number whose sine is which is called the principle value of sin. For this the only case is y. Therefore, for principal value of y sin, the domain is [, ] i.e. [, ] and range is y. Similarly, we can discuss the other inverse trigonometric functions. Domain Range (Principal value). y sin [, ],. y [, ] [0, ] 3. y tan R,. y cot R [0, ] 5. y sec or 0,, 6. y ec or, 0 0,. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS 8. Gr y sin y 8 MATHEMATICS

5 Inverse Trigonometric s MODULE - VII y tan y cot y sec y ec Fig.. Eample. Find the principal value of each of the following : (i) sin (ii) (iii) tan 3 Solution : (i) Let sin or sin sin or (ii) Let 3 3 or 3 (iii) Let tan 3 or 6 tan or tan tan 3 6 MATHEMATICS 9

6 MODULE - VII Inverse Trigonometric s Eample.3 Find the principal value of each of the following : (a) (i) (ii) tan ( ) (b) Find the value of the following using the principal value : sec 3 Solution : (a) (i) Let, then or (ii) Let tan ( ), then tan or tan tan (b) Let 3, then 3 or sec sec sec 6 3 Eample. Simplify the following : (i) sin Solution : (i) Let sin sin (ii) Let (ii) cot ec sin sin ec ec 30 MATHEMATICS

7 Inverse Trigonometric s Also cot ec MODULE - VII CHECK YOUR PROGRESS.. Find the principal value of each of the following : (a) 3 (d) (b) ec tan 3 (e) cot. Evaluate each of the following : (a) 3 (d) tan sec (e) (c) sin 3 (b) ec ec (c) ec 3 ec cot 3 3. Simplify each of the following epressions : (a) sec tan (d) cot (b) tan ec (e) tan sin (c) cot ec.3 PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS Property sin sin, Solution : Let sin Also sin sin sin sin sin Similarly, we can prove that (i), 0 (ii) tan tan, Property (i) ec sin (ii) cot tan MATHEMATICS 3

8 MODULE - VII (iii) sec Solution : (i) Let ec ec sin sin ec sin (iii) sec sec or sec Inverse Trigonometric s (ii) Let cot cot tan tan cot tan Property 3 (i) sin sin (ii) tan tan (iii) Solution : (i) Let sin sin or sin sin sin or sin or sin sin (ii) Let tan tan or tan tan tan or tan tan (iii) Let or Property. (i) sin (ii) (iii) ec sec tan cot 3 MATHEMATICS

9 Inverse Trigonometric s Soluton : (i) sin Let or (ii) Let or sin sin or sin cot cot tan tan cot tan (iii) Let ec ec sec sec ec sec Property 5 (ii) (i) or tan or sec y tan tan y tan y y tan tan y tan y Solution : (i) Let tan, tan y tan, y tan y We have to prove that tan tan y tan y By substituting that above values on L.H.S. and R.H.S., we have The result holds. Simiarly (ii) can be proved. L.H.S. = and R.H.S. tan tan tan tan tan tan tan L.H.S. MODULE - VII MATHEMATICS 33

10 MODULE - VII Inverse Trigonometric s Property 6 tan sin tan Let tan Substituting in (i), (ii), (iii), and (iv) we get tan tan tan...(i) sin tan sin tan sin sin sin sin...(ii) tan tan tan sin...(iii) tan tan From (i), (ii), (iii) and (iv), we get Property 7 tan tan tan...(iv) tan sin tan sin tan (i) sec (ii) cot sin tan ec Proof : Let sin (i), ec sin tan, sec cot sin tan sec, cot and ec 3 MATHEMATICS

11 Inverse Trigonometric s (ii) Let sin, and ec sec sin cot tan, ec sec, cot MODULE - VII tan ec sec Eample.5 Prove that tan tan tan Solution : Applying the formula : y tan tan y tan y, we have tan tan tan tan tan Eample.6 Prove that tan Solution : Let tan then tan tan L.H.S. = and R.H.S. = L.H.S. = R.H.S. Eample.7 Solve the equation tan tan, 0 MATHEMATICS 35

12 MODULE - VII Solution : Let tan, then tan tan tan tan tan Inverse Trigonometric s tan tan, tan 6 3, 3 6 Eample.8 Show that tan Solution : Let, then, Substituting in L.H.S. of the given equation, we have tan tan sin tan sin sin tan sin tan tan tan tan tan CHECK YOUR PROGRESS.. Evaluate each of the following : (a) sin sin 3 (b) cot tan cot (c) (d) y tan sin y tan tan 5 (e) tan tan 5 36 MATHEMATICS

13 Inverse Trigonometric s. If y, prove that y y sin 3. If y z, prove that y z yz. Prove each of the following : (a) (c) sin sin (b) tan tan (d) Solve the equation tan tan tan sin sin sin tan tan tan 5 8 MODULE - VII A C % + LET US SUM UP Inverse of a trigonometric function eists if we restrict the domain of it. (i) sin y if sin y where, y (ii) y if y = where, 0 y (iii) tan y if tan y = where R, y (iv) cot y if cot y = where R, 0 y (v) sec y if sec y = where, 0 y or, y (vi) ec y if ec y = where, 0 y or, y 0 Graphs of inverse trigonometric functions can be represented in the given intervals by interchanging the aes as in case of y = sin, etc. Properties : (i) sin sin, tan tan (ii), tan tan and ec sin, cot tan, sec sin sin (iii) sin sin, tan tan, (iv) (v) sin, tan cot, ec sec y tan tan y tan y, y tan tan y tan y (vi) tan sin tan MATHEMATICS 37

14 MODULE - VII (vii) sin tan sec cot ec Inverse Trigonometric s SUPPORTIVE WEB SITES TERMINAL EXERCISE. Prove each of the following : (a) (b) sin sin sin tan tan tan tan 5 5 (c). Prove each of the following : 3 tan tan tan 5 (a) (b) tan tan tan 3 (c) tan tan tan (a) Prove that sin sin (b) Prove that (c) Prove that sin 38 MATHEMATICS

15 Inverse Trigonometric s. Prove the following : (a) tan sin MODULE - VII (b) sin tan sin ab bc ca cot cot cot 0 a b b c c a (c) 5. Solve each of the following : (a) tan tan 3 (b) tan tan ec (c) sin 6 (d) cot cot, 0 MATHEMATICS 39

16 MODULE - VII ANSWERS CHECK YOUR PROGRESS. Inverse Trigonometric s. (a) 6 (b) (c) 3 (d) 3 (e). (a) 3 (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) CHECK YOUR PROGRESS.. (a) (b) 0 (c) y y 5. 0, TERMINAL EXERCISE (d) 5 (e) (a) 6 (b) (c) (d) 3 0 MATHEMATICS