Inverse trigonometric functions are defined as the inverse functions of the basic trigonometric functions, which are sine, cosine, tangent, cotangent, secant and cosecant functions. They are also termed arcus functions, antitrigonometric functions or cyclometric functions.
2
TRIGONO-
METRIC
FUNCTIONS
Syllabus Definition, range, domain, principal value branch, Graphs of inverse
trigonometric functions.
In this chapter you will study
Domain of inverse trigonometric functions.
Range of inverse trigonometric functions.
Principal branch value of inverse trigonometric functions.
Graph of different inverse trigonometric functions.
Revision Notes
As we have learnt in class XI, the domain and range of trigonometric functions are given below:
S. No. Function Domain Range
(i) sine R [– 1, 1]
(ii) cosine R [– 1, 1]
(iii) tangent { π
R − x : x = ( 2n + 1) ; n ∈ Z
2 } R
(iv) cosecant R – {x : x = np, n Î Z} R – (– 1, 1)
(v) secant { π
R − x : x = ( 2n + 1) ; n ∈ Z
2 } R – (– 1, 1)
(vi) cotangent R – {x : x = np, n Î Z} R
,
, 1. Inverse function
We know that if function f : X ® Y such that y =
f(x) is one-one and onto, then we define another
function g : Y ® X such that x = g(y), where x Î X
and y Î Y, which is also one-one and onto.
Key Words
One-one function: One to one
function or one to one mapping states
that each element of one set, say set A
is mapped with a unique element of
another set, say set B, where A and B
are two different sets.
In terms of function, it states as if f(x) =
f(y) Þ x = y, then f is one to one.
Onto function: If A and B are two
sets, if for every element of B, there is
atleast one or more element matching
with set A, it is called onto function.
Principal value branch of function cos–1: The graph
In such a case, Domain of g = Range of f
of the function cos–1 is as shown in figure. Domain
and Range of g = Domain of f of the function cos–1 is [–1, 1]. Its range in one of
g is called the inverse of f the intervals (– p, 0), (0, p), (p, 2p), etc. is one-one
g = f –1 and onto with the range [– 1, 1]. The branch with
or Inverse of g = g –1 = (f –1)–1 = f range (0, p) is called the principal value branch of
The graph of sine function is shown here: the function cos–1.
Thus, cos–1 : [– 1, 1] ® [0, p]
Principal value of branch function sin–1: It
is a function with domain [– 1, 1] and range
−3 π − π − π π π 3π
2 , 2 , 2 , 2 or , and so on
2 2
corresponding to each interval,
we get a branch of the function
sin– 1
x. The branch with range
−π π
2 , 2 is called the principal
value branch. Thus, sin–1 : [–1, 1]
−π π
® , .
2 2
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