Comment:

When illustrating to students the connection between the unit circle and its map into a sine or cosine function, one could say that the distance of the wave line y=sinx is the radian value of x.

— So, the distance from the start of the sine function to the end of its first iteration at 2pi (360) is 2pi. It would seem so, but do not have any formal proof for it.

]]>1. Trigonometric libraries are slowed down when calculating large numbers, where cycles could make it trivial to perform the calculation: simply drop the whole number, and do the calculation on the whole number part.

2. 360 divides nicely by factors of 10, 9, 8. This means the degree measure for any of those angles can be stored exactly as a finite binary number.

As for Euler’s formula, cos t + j sin t = e^jt is kind of nice, but when you start doing Fourier transforms, you see e^(j*2*pi…) terms, and the reason the 2*pi is there is because you’re using radians instead of cycles. Use cycles, and the term goes away.The Fourier transform (along with its variants like cosine transform) is widely used in engineering and science for everything from jpeg image compression to radio frequency modulation. A silly thing that slows down trig functions in FFT calculation is the common use of radians for cosine and sine in standard libraries where it’s completely unnecessary.

]]>A much more secure way is to drill the hole in the brass screw slightly smaller than the diameter of the steel shaft and then heat it to point where it is larger than the steel diameter. Then quickly push it over the the steel shaft until it cools. Is this practical.? ]]>

Similar thing that has always dissatisfied me is how repeated roots are handled in the characteristic equation for ODEs. The method basically boils down to “here’s the answer, check it it works.” Which is a little bit like saying “this is the integral, just differentiate and you will see”. Sure that shows the answer is correct. But it doesn’t help in deriving it. The one book that I have found that did a nice job of a natural derivation was actually 4th edition Thomas Finney Calculus.

Would note on your problem above that a lot of books give a general method with a subsitution of half angle tangent. This works in general for sin cos type problems. And the sec problem yields to it. Even here would like a little intuitive rationale for why it is the key to unlock so much.

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