Mathematics : Inverse Trigonometric Functions: Definition and Some Fundamental Concepts

**Some Fundamental Concepts**

A real valued function *f *is **periodic **if there exists a
number *p *> 0 such that for all *x *in the domain of *f *, *x
*+ *p *is in the domain of *f *and *f *(*x *+ *p*)
= *f *(*x*) .

The smallest of all such numbers, is called the **period **of the function *f *.

For instance, sin *x*, cos *x*, cosec*x *, sec *x
*and *e*^{ix}* *are periodic functions with period 2*π *radians, whereas tan *x
*and cot *x *are periodic functions with period *π *radians.

A real valued function *f *is an **even function **if for all *x *in
the domain of *f *, −*x *is also in the domain of *f *and *f *(−*x*)
= *f *(*x*) .

A real valued function *f *is an **odd function **if for all *x *in
the domain of *f *, −*x *is also in the domain of *f *and *f
*(−*x*) = − *f *(*x*) .

For instance, *x*^{3} , sin *x*, cosec *x*, tan *x *and
cot *x *are all **odd functions****, **whereas *x*^{2}, cos *x *and sec *x *are **even functions**.

**Remark**

The period of *f* = g ± h is lcm{period of g, period of h }
, whenever they exist.

For instance, the period of *y *= cos 6*x *+ sin 4*x
*is *π *and that of *y *= cos *x *- sin *x *is 2* π*

The **domain **and **range **of trigonometric functions are given in the following table.

Let *f *: **R** → R be a real valued function and *f *(*x*)
be the value of the function *f *at a point *x *in the domain. Then,
the set of all points ( *x*, *f *(*x*)), *x *∈ R determines the graph of the function *f *. In general, a graph in *xy
*-plane need not represent a function. However, if the graph passes the
vertical line test (any vertical line intersects the graph, if it does, atmost
at one point), then the graph represents a function. A best way to study a
function is to draw its graph and analyse its properties through the graph.

Every day, we come across many phenomena like tides, day or night
cycle, which involve periodicity over time. Since trigonometric functions are
periodic, such phenomena can be studied through trigonometric functions. Making
a visual representation of a trigonometric function, in the form of a graph,
can help us to analyse the properties of phenomena involving periodicities.

To graph the trigonometric functions in the *xy *-plane, we
use the symbol *x *for the independent variable representing an angle
measure in radians, and *y *for the dependent variable. We write y =
sin x to represent the sine function, and in a similar way for other
trigonometric functions. In the following sections, we discuss how to draw the
graphs of trigonometric functions and inverse trigonometric functions and study
their properties.

The amplitude is the maximum distance of the graph from the *x
*-axis. Thus, the amplitude of a function is the height from the *x *-axis
to its maximum or minimum. The period is the distance required for the function
to complete one full cycle.

(i) The graph of a periodic function consists of repetitions of
the portion of the graph on an interval of length of its period.

(ii) The graph of an odd function is symmetric with respect to
the origin and the graph of an even function is symmetric about the *y *-axis.

Remember that a function is a rule that, given one value, always
gives back a unique value as its answer. For existence, the inverse of a
function has to satisfy the above functional requirement. Let us explain this
with the help of an example.

Let us consider a set of all human beings not containing identical
twins. Every human being from our set, has a blood type and a DNA sequence.
These are functions, where a person is the input and the output is blood type
or DNA sequence. We know that many people have the same blood type but DNA
sequence is unique to each individual. Can we map backwards? For instance, if
you know the blood type, do you know specifically which person it came from?
The answer is NO. On the other hand, if you know a DNA sequence, a unique
individual from our set corresponds to the known DNA sequence. When a function
is one-to-one, like the DNA example, then mapping backward is possible. The
reverse mapping is called the inverse function. Roughly speaking, the inverse
function undoes what the function does.

For any right triangle, given one acute angle and the length of
one side, we figure out what the other angles and sides are. But, if we are
given only two sides of a right triangle, we need a procedure that leads us
from a ratio of sides to an angle. This is where the notion of an inverse to a
trigonometric

function comes into play.

We know that none of the trigonometric functions is one-to-one
over its entire domain. For instance , given sin*θ *= 0.5 , we have
infinitely many *θ *= satisfying the equation. Thus, given
sin*θ *, it is not possible to recover *θ *uniquely. To overcome the
problem of having multiple angles mapping to the same value, we will restrict
our domain suitably before defining the inverse trigonometric function.

To construct the inverse of a trigonometric function, we take an
interval small enough such that the function is one-to-one in the restricted
interval, but the range of the function restricted to that interval is the
whole range. In this chapter, we define the inverses of trigonometric functions
with their restricted domains.

Assume that *f *is a bijective function and *f *^{−1} is the inverse of
*f *. Then, *y *= *f *(*x*) if and only if *x
*= *f*^{ }^{−1} ( *y*) . Therefore, (*a*, *b*) is a point on the
graph of *f *if and only if (*b*, *a*) is the corresponding
point on the graph of *f*^{−1} . This suggests that graph of the inverse
function *f *^{−1} is obtained from the graph of *f *by interchanging *x
*and *y *axes. In other words, the graph of *f *^{−1} is the mirror image of the graph of *f *in
the line *y *= *x *or equivalently, the graph of *f *^{−1} is the reflection of
the graph of *f *in the line *y *= *x*.

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12th Mathematics : UNIT 4 : Inverse Trigonometric Functions : Some Fundamental Concepts of Inverse Trigonometric Functions |

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