Euler's Formula of Inverse Trig Functions

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There is no clean exponential formula for inverse trigonometric functions like arcsine and arccosine, similar to the formulas for sine and cosine. While these functions can be expressed using logarithms and square roots, they do not have a straightforward exponential representation. The process involves manipulating the sine function's formula in terms of exponentials and then solving for the exponential term, which leads to a quadratic equation. After solving, logarithms can be applied, but the result remains complex. Ultimately, the simplicity found in sine and cosine does not extend to their inverse counterparts.
Tac-Tics
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Is there a clean exponential formula for inverse trig functions (arcsine, arccosine) like there is for sine and cosine?
 
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Tac-Tics said:
Is there a clean exponential formula for inverse trig functions (arcsine, arccosine) like there is for sine and cosine?

Short answer … no! :redface:

Long answer … noooooooo! :cry:

You knew that didn't you? :smile:​
 
tiny-tim said:
Short answer … no! :redface:

Long answer … noooooooo! :cry:

You knew that didn't you? :smile:​

I figured as much. Mathematics is as cruel as she is beautiful.
 
You can write them with logarithms and square roots: first write the formula for, say, sin(z) in terms of exp(iz) and exp(-iz). Then solve for exp(iz) (it's a quadratic equation). Then take logarithms.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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