Introduction toInverse Trig Functions: Let us recall that the inverse of a function f: A -> B exists if and only if f is one-one-onto i.e., a bi-jection and is given byf(x) = y=>f^-1(y) = x. Often it happens that a given function may not be one-one in the whole of its domain but when we restrict it to a part of the domain, it becomes one-one. If a function is one-one on a part of its domain, it is said to be invertible on that part only. If a function is invertible in several parts of its domain, it is said to have an inverse in each of these parts. Let us understand Inverse Trig Functions and we will learn
Graphing Inverse Trig Functions : Inverse sine function: The inverse of sine function is defined assin^-1 : [-1, 1] -> [-pi/2, pi/2] such thatsin^-1 x = Theta and sin (theta) = x. The graph of sin^-1x can be obtained by interchanging x and y axes in the graph of y = sinx .Thus if (a, b) is a point on the graph of y = sinx , then the point (b , a) becomes the corresponding part on the graph of y = sin^-1x , we can also observe from the graph of inverse sin function that graph of y = sin^-1x with domain (-1, 1) and range [-pi/2, pi/2] Inverse cosine function: The branch with range [0, pi] is the principal value branch.
Thus inverse of cosine function is cos^-1 : [-1, 1] -> [0, pi].The graph of y = cos^-1x is not symmetric about the origin. The graph of cos^-1x can be obtain by reflection across the line y= x is a portion of the curve x = cosy. Inverse cosecant function: The function corresponding to the range [-pi/2, pi/2] – {0} is the principal value branch of cosec^-1. Thus the inverse of cosecant function is given byCosec^-1 : R - ] -1, 1 [ -> [-pi/2, pi/2] – {0}.Graph of y = cosec^-1 x have Domain: ] –infinity, -1] union [1, infinity[ and range is [-pi/2, 0 [ union]0, pi/2]Inverse secant function: The branch with range [0, pi] – {pi/2} is called principal value branch of the inverse secant function.
Thus the inverse of the secant function issec^-1 : R - ] -1, 1 [ -> [0, pi] –{pi/2}.Graph of y = sec^-1 x have Domain ] –infinity, -1] union [ 1, infinity [ and range [0, pi/2 [ union ] pi/2, pi Inverse tangent function :The branch with range]-pi/2, pi/2[ is the principal value branch of inverse tangent function. Thus the inverse of the tangent function istan^-1 : R -> ]-pi/2, pi/2[.The graph of y= tan^-1x is symmetric about the origin because it is a branch of the graph x = tan y that is symmetric about origin.
Algebraically the arc tangent is an odd function n Inverse Cotangent function: The graph of y = cot ^-1x has no such symmetry .
Let us understand inverse of trig function through an example of Solving inverse trig functions :
suppose we have to solve y = sin^-1 (1/2) for solving inverse trig functionsfirst we will sin y = ½.As sine of an angle is positive in 1st and 2ndquadrants,sin y =1/2 = sin (pi/6) orsin (pi – pi/6)=> sin (pi/6)r sin(5pi/6) or Y = pi/6or 5pi/6.Hence y = pi/6 and y = 5pi/6 satisfy y = sin^-1 (1/2).Integrals of Inverse
Trig Functions: Let us start with the integral of inverse sin function . The integral of (sin^-1 x) dx = x sin^-1 x +sqrt(1 – x^2) + c . The integral of Integral (tan^-1 x) dx = x tan^-1 x – ½ log | 1 + x^2| + c. and the Integral (sec^-1 x) dx = x sec^-1 x – log |x + sqrt(x^2 – 1)| + c.
