Inverse Trigonometric Functions

Consider a function f(x)=(x+2), we can write this as y=x+2. Solving for x, we get, x = y-2. Interchanging the x and y terms gives a new function y=x-2, consider this as g(x). This new function g(x) is the inverse function of f(x). Inverse function is denoted as f^-1

Inverse Trigonometric Functions
Let us assume a given radian angle pi/2, we can evaluate a trigonometric function sin(pi/2)  equals 1. Inversely, if we are given 1 as the value of a sine function, then we can arrive to the radian angle y. Equating,  sin(pi/2) = 1, the radian angle for which the value of sine is 1 is pi/2. We can now write the inverse trigonometric function of sine as arc[sin(1)]= pi/2. In general, given the value x of the trigonometric function we can write the inverse trigonometric function as arc[sin(x)], arc[cos(x)],arc[tan(x)], arc[csc(x)], arc[sec(x)], arc[cot(x)] to name the radian angle that has the value x in each case.

Using the derivatives of inverse trigonometric identities we can obtain inverse trig functions integrals:

Integral[du/sqrt(a^2-u^2)] = sin^-1[u/a] +c
Integral[du/(a^2+u^2)=1/a[tan^-1(u/a)]+c
Integral[du/u[sqrt(u^2-a^2)]=1/a[sec^-1mod(u/a)] +c;
where u is a function of x, which is written as u=f(x)

How to Solve Inverse Trig Functions
Steps involved to solve inverse trig function, arc tan(-1):
Step1:  let y=arc tan(-1)
Step2:  tan (y) is -1 when pi/2<=y<=pi/2 [finding the interval]
Step3:  determine the radian angle of tan that gives the value of 1, which is pi/4
Step4:  determine the radian angle for a negative value, we know that tan(-x)=- tan(x)
Step5:  we get, tan y = -1
Step6:  the required radian angle is y = pi/4

Inverse Trig Functions Problems
Evaluate arc cos(-1/2)
Let y = arc cos(-1/2), we need to find the radian angle y
cos(y) = -1/2 , we know that pi/3 radian angle of cos gives the value 1/2. But here we have -1/2
We know that, cos(pi-x) gives us –cosx.
So, cos(pi-pi/3) = -1/2
We get, y=pi – pi/3 = 2pi/3 is the required radian angle

How to Graph Inverse Trig Functions
To graph inverse trig functions, first we need to understand the domain and range of the inverse trig functions which are:
Function Domain Range
y=arc sin(x)  -1<= x<= 1    -(pi/2)<=y<=(pi/2)
y=arcos(x)   -1<= x<=1   0<=y<=pi
y=arctan(x)   -infinity
y=arccsc(x)  x<=-1or x>=1  -pi/2<=y<=pi/2, y not equals zero
y=arcsec(x)   x<=-1 or x>=1  0<=y<=pi, y not equals pi/2
y=arccot(x)      -infinity  0

Plug in different values of x (within the range) to arrive to the respective y values of the given inverse trig function. Plot the coordinates (x,y) on the graph and join them. The curves thus obtained is the graph of given inverse trig function.