4 ms·
wait what? ELI grad student, I know you can plug i into the sine taylor series expansion but what is happening here
by dustingetz 4y ago
wait what? ELI grad student, I know you can plug i into the sine taylor series expansion but what is happening here
- some_furry 4y agoThis is radians, which can go from 0 to tau (2 * pi, about 6.28).
- deleted 4y ago[deleted]
- scythe 4y agosin(x + pi/2) = cos(x) = cosh(ix) Therefore, sin (pi/2 + i arcosh(2)) = cosh(-arcosh(2)) = 2 where arcosh(y) is the familiar inverse of the hyperbolic cosine, i.e. log(y + sqrt(y^2 - 1)), so we find sin(pi/2 + i log(2 + sqrt(3))) = 2.
- hansvm 4y agoSin/cos are tightly connected to the complex exponential and exhibit a lot of such behavior. In particular, sin(x+it) = sin(x)cosh(y) + i cos(x)sinh(y). Set x something to zero out cosine, and rely on the exponential nature of cosh to find a target y value.
- hansvm 4y agoOops, autocorrect, sin(x+iy). Also x and y should be real.
- version_five 4y agoThe above should be intelligible to someone with high school math, no? I'd be interested in the grad student version, just out of curiosity
- nroets 4y agoMost highschool students mainly use sin to calculate the lengths of sides of triangles. So it feels very counterintuitive that sin can be greater than 1. If you define sin as the solution to a particular differential equation, then it's relationship to the hyperbolic functions is much more obvious.