Inverse Trigonometric Functions. Inverse of a function f eists, if function is one-one and onto, i.e., bijective.. Trignometric functions are many-one functions but these become one-one, onto, if we restrict the domain of trignometric functions. We can say that inverse of trigonometric functions are defined within restricted domains of corresponding trigonometric functions.. Inverse of sin is denoted by sin (arc sin function). We also write as sin. Similary, other inverse trigonometric functions are given by cos, tan,sec,cot and. Note that sin sin and (sin ) sin. Also sin (sin ). 5. Table for domain and range of inverse trigonometrical functions : cosec. Functions Domain Range (Principal value Branch) y = sin y y = cos 0 y y = tan < < < y < y = cosec or y, y 0 y = sec or 0 y, y y = cot < < 0 < y < Properties of Inverse Trigonometric Functions. (i) sin (sin θ ) = θ, for all θ [, ] cos (cos θ ) = θ, for all θ [0, ] tan (tan θ ) = θ, for all θ [, ] (iv) cosec (cosec θ ) = θ, for all θ [, ], θ 0 (v) sec (sec θ ) = θ, for all θ [ θ, ], θ /
(vi) cot (cot ) θ = θ, for all θ [0, ]. (i) sin(sin ) =, for all [, ] cos(cos ) =, for all [, ] tan(tan ) =, for all R (iv) cosec(cosec ) =, for all (, ] [, ] (v) sec(sec ) =, for all (, ] [, ] (vi). (i) cot(cot ) sin =, for all R = cosec, for all (, ] [, ] = cos sec tan > + <, for all (, ] [, ] cot, for 0 cot, for 0. (i) sin ( ) = sin ( ), for all [, ] cos ( ) cos, = for all [, ] tan ( ) = tan,, for all R (iv) cosec ( ) = cosec,, for all (, ] [, ] (v) sec ( ) = sec,, for all (, ] [, ] (vi) cot ( ) = cot, for all R 5. (i) sin cos / + =, for all [, ] tan cot / + =, for all R sec + cosec = /, for all (, ] [, ] 6. (i) sin + sin y = { } y y sin +, if, y and + y or if y < 0 and + y > sin sin y { y y } = sin, if, y and + y or if y < 0 and + y > 7. (i) cos + cos y = { } y y cos, if, y and + y 0
cos cos y = { } y y cos, if, y and + y 0 y 8. (i) tan tan y tan + + = y, if y < y tan tan y tan, = + y if y > 9. (i) sin = sin ( ), if cos = cos ( ), if 0 tan = tan, if < < 0. (i) sin = sin ( ), if cos = cos ( ), if tan = tan, if < <. (i) + tan = sin, tan = cos, + if if 0 <. (i) sin = cos = tan = cot = sec cosec = cos = sin = tan = cot sec cosec = =
tan = sin cos = + + = cot sec cosec + = + = Important substitution to simplify trigonometrical epressions involving inverse trigonometric functions : Epression Substitution a a + = a tan θ or = a cot θ = a sin θ or = a cosθ a = a secθ or = a cosec θ a + a or a a + = a cos θ Important Facts (i) If no branch of an inverse trigonometric function is mentioned, we mean the principal value branch of that function. sin sin or (sin ) and same holds true for other trigonometric functions also. If sin y sin = then and y are the elements of domain and range of principal value branch of respectively. i.e., [, ] and y,, Similar fact is also applicable for other inverse trigonometric functions. Question for Practice Very Short Answer Type Questions ( Mark). Evaluate : sin sin. Using principal value, evaluate : cos cos + sin sin
. Show that sin ( ) = sin.. Solve for = > + :tan tan, 0. 5. What is the principal value of tan ( )? 6. What is the principal value of 7. Using principal value evaluate cos? sin sin 5. 8. Write the principal value of 9. Write the principal value of cos cos 6. 7 tan tan. 0. What is principal value of sin.. What is domain of the function sin. Write principal value of sec ( ).?. Write the principal value of cos sin.. Find the principal value of tan sec ( ). 5. Write the value of cot(tan a + cot a). 6. Write the principal value of tan () + cos. 7. Write the value of tan tan. 5 8. Write the principal vlaue of tan ( ) cot ( ). 9. Write the value of tan sin cos. 0. Write the principal value of 9 tan tan. 8
. Write the value of sin sin. 5 Very Short Answer Type Questions ( Mark). Prove that : tan + cos + tan cos = b b a a b. a. Solve for 8 : tan ( + ) + tan ( ) = tan.. Prove that : tan + tan = tan. 5 8 7. Prove that : 5. Prove that : 6 sin + cos + tan. 5 6 tan + tan + tan + tan =. 5 7 8 6. Solve for 7. Solve for + = : tan tan. :tan tan. + + = + 8. Prove that : tan tan tan + = 7 7. 9. Prove that : tan + tan + tan =. 5 8 0. Prove that : cot (7) + cot (8) + cot (8) = cot ().. Prove that : + = + < < tan tan,0.. Solve for = > + :tan tan 0, 0.. Prove that :. Prove that : 5 6 sin + sin + sin. 5 5 tan + tan = cos 9 5.
sin sin 5. Prove that : cot + + =, 0,. + sin sin 6. Solve for : tan (cos ) = tan (cosec ). 7. Solve for + = :cos tan. + 8. Prove that : + tan = cos, (0,). 56 Prove that : cos + sin = sin 5 65. 9. Prove that following : a a b + b + = b a tan cos tan cos. 0. Prove that following : Prove the following : tan + tan = tan. + cos[tan {sin(cot )}] =. + sin sin. Prove the following : cot + + =, 0,. + sin sin Find the value of y tan tan +. y + y. Prove the following : 9 9 9 = 8 sin sin. Solve the following equation for. tan = tan, > 0. +. Prove that : + tan = cos,. + +
cos r. Prove that : tan r r =,,. sin 8 5 Prove that : sin + sin = cos. 7 5 85 6 5. Prove the following : cos sin + cot =. 5 5 6 5 6. Prove that : sin = sin + cos. 65 5 Solve for : tan (sin ) = tan (sec ),. y 7. Find the value of the following : tan sin cos,, y 0 and y. + < > < + + y Prove that : tan + tan + tan =. 5 8 7 8. Show that : tan sin =. Solve the following equation : cos(tan ) = sin cot. 9. If cos(tan ) = sin cot, then find the value of. If y = cot ( cos ) tan ( cos ), then prove that sin y = tan. ANSWERS Very Short Answer Questions :... 5. 6. 6 7. 5 8. 5 6
9.. 7.. 0. 5 5 Short Answer Questions :. 8.. [, ]. 5. 0 6. 9. 0. 8. 6. 6 7. ±. 6. 7.,.. 6. 7. + y y 8. = 9. =±