Inverse Trigonometric Functions
Inverse Trigonometric Functions
Before we study about inverse trig functions, it is important for us to know some of the
basics about trigonometry.
The word trigonometry itself defines its meaning as the first part of the word
trigonometry is "trigon" which has a meaning "triangle" whereas the second part of the
word is "metron" which has a meaning "measuring".
So trigonometry is used to measure the elements: sides and angles, of a triangle.
There are enormous number of trigonometric inequalities and equations in
trigonometry.
And if we consider the modern time, then we have six trigonometric functions: sine,
cosine, tangent, secant, cosecant and cotangent. Out of these six the last three are
derived from the first three functions.
Secant is the trigonometric function which is the reciprocal of the function cosine.
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Cosecant is the reciprocal of the trigonometric function sine and the last one
cotangent is the reciprocal of the trigonometric function tangent.
And tangent can also be represented as the ratio of two trigonometric functions which
are sine and cosine. All these formulae are also true for inverse trig functions.
There is one another term in trigonometry, which is inverse trig functions or
occasionally called cyclometric functions are nothing but the inverse functions of the
trigonometric functions with different restricted domains.
We use notations for inverse trig functions and they are: sin-1, cos-1, tan-1, cosec-1,
sec-1 and cot-1 , and they are often used as arc(sin), arc(cos), arc(tan), arc(cosec),
arc(sec) and arc(cot).
But when we represent inverse trig functions as sin-1 ,then this convention might
create some conflicts with the common multiplication expression of trigonometric
functions like sin2(x), which means a numeric power and not a function composition.
So this might create a confusion between multiplication inverse and composition
inverse, therefore we usually use arc(sin) for inverse trig functions.
One important thing to be noted is that none of the trigonometric functions is onto
functions, therefore they must have separate restricted domains so as to have inverse
trig functions.
In computers, the inverse trig functions arc(sin), arc(cos), arc(tan), arc(sec), arc(cot)
and arc(cosec) are often represented as asin, acos, atan, asec, acosec and acot.
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Inverse trig functions are bounded in ranges which are subsets of the domains of the
parent trigonometric functions.
The inverse trig functions like arcsin(x), arcsoc(x) are assumed to be equal to some
number like arcsin(x) = y such that sin(y) = x and to define ranges to the inverse trig
functions, we test multiple numbers of y for which sin(y) = x; for instance, let us start
with zero, sin(0) = 0, but for every n value of y it is 0, sin() = 0, sin(2) = 0, etc.
It shows that the inverse trig functions are multivalued functions like arcsin(0) = 0,
arcsin(0) = and also arcsin(0) = 2 and so on till n value of y.
But when we need only single value, the inverse trig functions are restricted to its
domain. When we apply such boundations on inverse trig functions, then for each
value of x which should be in the domain, arcsin(x) will be solved to a single value
only which will be termed as its principal value. Every inverse trig function has its own
domain.
Like for arcsin x has domain [-1, 1], arcos has [-1, 1], in arctan x can be any real
number and so on.


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