Inverse Trigonometric Functions

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 nπ value of y it is 0, sin(π) = 0, sin(2π) = 0, etc.

It shows that the inverse trig functions are multivalued functions like arcsin(0) = 0, arcsin(0) = π and also arcsin(0) = 2π and so on till nπ 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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