Explaining the derivatives of sin, cos and toa?

I am assuming this isn’t homework, as it is summer and from what you wrote it appears you are merely curious, so I think it’s legal for me to do the proof here for you (don’t strike me down, Fluther gods!)

So the limit definition of a derivative is lim h->0 (f(x+h)-f(x))/h.
Let’s substitute the sin function in for f(x).
limit h->0 (sin(x+h) – sin(x))/h
A trigonometric identity says that sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
So let’s rewrite our limit using that:
limit h->0 (sin(x)cos(h) + cos(x)sin(h) – sin(x))/h
Rearrange that by removing common factors:
limit h->0 (sin(x)(cos(h) – 1) + cos(x)sin(h))/h
limit h->0 sin(x)(cos(h)-1)/h + cos(x)(sin(h)/h)
The limit as h->0 of (cos(h)-1)/h is 0, and of sin(h)/h is 1 (You may already know why this is so I won’t prove it here, but if you do not know where I got that from, just let me know and I’ll be happy to do those proofs out for you too).
Which leaves you with sin(x)(0) + cos(x)(1) or just cos(x).

Cos is calculated very similarly; let me know if you’d like to see the proof for that as well.
By toa do you mean tangent? I can prove that one too if you’d like.

Hey, just checking back. Did you want to see any of the other proofs?

There’s also a geometric interpretation: the derivative is the slope of the tangent line. So in the case of the sine function, it starts with maximum up-slope at the origin (deriv. = 1) and then levels off to horizontal at pi/2 (deriv. = 0), then maximum down-slope at pi (deriv. = -1) then levels off again at 3pi/2, then horizontal (slope=0) again at 2pi. So the slope of the sine curve has a value that exactly follows the cosine function, confirming d(sin x)/dx = cos x. Not an actual derivation as @Mariah gave, but might help you remember.

Similarly d(cos x)/dx = – sin x.
Derivative of tangent d(tan x)/dx = (1 / cos^2) unfortunately not as intuitive.

Once you have the derivative of sin(x)=cos(x), it is easy to get the derivative of cosine, using the relationship cos(x) = sin(pi/2 – x) and sin(x) = cos(pi/2 – x)

Derivative of cos(x) = derivative of sin(pi/2 – x) = -cos(pi/2 – x) = -sin(x)

@Mariah
That was great