A Note . sin – x should not be confused with (sin x ) – . In fact (sin x ) – = sin x and similarly for other trigonometric functions. .
Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function. . The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions. We now consider some examples: Example Find the principal value of sin – .
Solution Let sin – = y . Then, sin y = . We know that the range of the principal value branch of sin – is and sin . Therefore, principal value of sin – is Example Find the principal value of cot – Solution Let cot – = y .
Then, cot y = − = cot π − = We know that the range of principal value branch of cot – is ( , π ) and cot − . Hence, principal value of cot – is EXERCISE . Find the principal values of the following: . sin – .
cos – . cosec – ( ) . tan – ( ) . cos – .
tan – (– ) INVERSE TRIGONOMETRIC FUNCTIONS . sec – . cot – ( ) . cos – .
cosec – ( Find the values of the following: . tan – ( ) + cos – + sin – . cos – + sin – . If sin – x = y , then (A) ≤ y ≤ π (B) y (C) < y < π (D) y .
tan – ( sec is equal to (A) π (B) (C) (D) . Properties of Inverse Trigonometric Functions In this section, we shall prove some important properties of inverse trigonometric functions. It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions.
In fact, they will be valid only for some values of x for which inverse trigonometric functions are defined. We will not go into the details of these values of x in the domain as this discussion goes beyond the scope of this textbook. Let us recall that if y = sin – x , then x = sin y and if x = sin y , then y = sin – x . This is equivalent to sin (sin – x ) = x , x ∈ [– , ] and sin – (sin x ) = x , x ∈ For suitable values of domain similar results follow for remaining trigonometric functions.
We now consider some examples. Example Show that (i)