Related (duplicate?): Simple proof of Euler Identity $\exp i\theta = \cos\theta+i\sin\theta$. Also, this possible duplicate has this answer, with a nice visual demonstration of the result. There are more instances of this question floating around Math.SE. Try searching for variations of "euler identity proof"; if no existing answers satisfy you, try to convey what it is about them that you ...
What is the geometrical importance of the Euler Line (ie, the line through the centroid, orthocenter, and circumcenter (and other points) of a non-equilateral triangle)? What is meant by importanc...
It was found by mathematician Leonhard Euler. In 1879, mathematician J.J.Sylvester coined the term 'totient' function. What is the meaning of the word 'totient' in the context? Why was the name coined for the function? I have received replies that 'tot' refers to 'that many, so many' in Latin. What about the suffix 'ient'?
Euler's formula is quite a fundamental result, and we never know where it could have been used. I don't expect one to know the proof of every dependent theorem of a given result.
The $3$ Euler angles (usually denoted by $\alpha, \beta$ and $\gamma$) are often used to represent the current orientation of an aircraft. Starting from the "parked on the ground with nose pointed North" orientation of the aircraft, we can apply rotations in the Z-X'-Z'' order: Yaw around the aircraft's Z axis by $ \alpha $ Roll around the aircraft's new X' axis by $ \beta $ Yaw (again) around ...
3. How Euler Did It This is just a paraphrasing of some of How Euler Did It by Ed Sandifer - in particular, the parts where he paraphrases from Euler's Introductio. Note that Euler's work was in Latin, used different variables, and did not have modern concepts of infinity. I'll use $\mathrm {cis}\theta$ to denote $\cos\theta+i\sin\theta$.
0 There is one difference that arises in solving Euler's identity for standard trigonometric functions and hyperbolic trigonometric functions. The difference is that the imaginary component does not exist in the solution to the hyperbolic trigonometric function.
Well, the Euler class exists as an obstruction, as with most of these cohomology classes. It measures "how twisted" the vector bundle is, which is detected by a failure to be able to coherently choose "polar coordinates" on trivializations of the vector bundle.
Euler rates are confusing. Could you provide more specific clarification? e.g. you first say we're transforming from body-frame angular velocity to ZYX Euler rates (body-frame as well?) then later say we're rotating about an inertial frame. Also could you specify if Z in ZYX corresponds to $\phi$ which corresponds to roll or yaw?
