Prove that d/dx sin(x)=cos(x) iff the period of sin(x)=2pi

Question

debbieg · Answer

Find $$\large \frac{ d }{dx }\sin(bx)$$
Which will involve the chain rule and be a function of b.

What must b= in order for that that derivative to =cos(x)?

When b= that, what is the period of sin(bx)?

mathsolver · Answer

d/dx sin(bx)=bcos(bx)
therefore b=1

debbieg · Answer

Right, any other value for b and the derivative is not =cos(x).

Although, any other value for b, and you aren't taking the derivative of sin(x), lol... you would take taking the derivative of sin(bx) which is not =sin(x) if b is not=1. So it seems like a bit of an oddly worded problem.  :)

mathsolver · Answer

What I mean is why do we use radians for trigonometric functions. I know it means we don't have to mess around with coefficients when we differentiate. But how do we know it's radians that has this nice property

debbieg · Answer

Gosh, I'm sorry, I guess I don't really understand what you're asking. But my simplistic answer would be, we use radians because radians are just real numbers, and hence all the properties and operations we apply to them can be carried out without any kind of conversions....?

Don't ask "why use radians", ask, "why NOT use radians?" ;) lol

mathsolver · Answer

Haha, yeah, fair enough. I just want to know how we know that radians have this lovely property but degrees don't

debbieg · Answer

The property being that $\large \dfrac{ d }{dx }\sin(x)=\cos(x)$?