Inverse trigonometric functions

Transformations of inverse trigonometric functions need to be handled with care. Consider the identity sin2theta=(2tantheta)/(1+tan^2theta) in its domain of definition. Suppose we set tantheta=x , we have sin2theta=(2x)/(1+x^2) Taking sin^(-1) of both sides yields 2theta=sin^(-1).(2x)/1+x^2i.e.,2tan^(-1)x=sin^(-1).(2x)/(1+ x^2) . But we will discover that the above identity is not valid for all x. Choose x=sqrt3, LHS =2tan^(-1)sqrt3=2xxpi/3=(2pi)/3, RHS=sin.(2sqrt3)/(1+3)=sin^(-1).(sqrt3)/2=pi/3 . And so left hand and right hand side don't match. The reason is that we have disregarded the principal values of inverse functions. So it is well to remember that the iverse trigonometric formmulae have restrictions attached to the argument. When the values of x lie outside the interval of validity then the formula needs to be corrected. Let f(x)=sin^(-1)(x)/(1+x^2),g(x)=2tan^(-1)x . Then the largest interval in R on which f and g both are agree

Some Important properties of Inverse Trigonometric function

The graph of inverse trigonometric function can be obtained from the graph of their corresponding function by interchanging X and Y-axes.

Write tan^(-1) x, x gt 0 in the form of other inverse trigonometric function

If int(dx)/(x^2+a x+1)=f(g(x))+c , then f(x) is inverse trigonometric function for |a|>2 f(x) is logarithmic function for |a| 2 g(x) is rational function for |a|<2

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)

For any positive integer n , define f_n :(0,oo)rarrR as f_n(x)=sum_(j=1)^ntan^(-1)(1/(1+(x+j)(x+j-1))) for all x in (0, oo) . Here, the inverse trigonometric function tan^(-1)x assumes values in (-pi/2,pi/2)dot Then, which of the following statement(s) is (are) TRUE? sum_(j=1)^5tan^2(f_j(0))=55 (b) sum_(j=1)^(10)(1+fj '(0))sec^2(f_j(0))=10 (c) For any fixed positive integer n , (lim)_(xrarroo)tan(f_n(x))=1/n (d) For any fixed positive integer n , (lim)_(xrarroo)sec^2(f_n(x))=1

The least numerical value, either positive or negative of angle theta is called principal value of the inverse trigonometric function.

Find the values of the other five trigonometric functions I each of the following: cottheta=(12)/5,theta in quadrant III

Find the values of the other five trigonometric functions in each of the following: costheta=-1/2,theta in quadrant II