The basic idea behind this wiki is to demonstrate that how we can evaluate inverse trigonometric functions outside their domain using Complex Analysis And Euler's Formula.
Everyone of us knows that the range of circular functions : and is .
So did you ever tried solving the equation or in particular, finding the solutions for :
Yeah, you guessed it right, the solution is bit too complex!!!
Since, we are solving an equation, for such a point which is outside the range of a function, we know that there won't exist a real solution.
So, how should we start...?
In fact, because we're dealing with both complex numbers and trigonometric functions, that gives us a clue of starting with the Euler's Identity :
Today, I am going to introduce you to a method with which you can easily evaluate and for all real value of .
So, here we go -
First of all, by Euler Formula, we have :
Subtracting them up, gives
Now, we wish to find the solutions for . So,
So, now can you look up the quadratic coming around...?
No! ok, I'll just a use a simple substitution here, which makes the work tidier and easier to see the quadratic.
Substituting , we get
Upon Rearranging, we get
Now, that's a quadratic in whose solutions are -
Taking natural logarithm and multiplying by on both and yields - And thus,
Since, so lies in the first and fourth quadrants, and hence we do not need to make any changes in our formula.
1. Evaluate : 2. Solve for :