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The derivative of the sine of angle is the cosine of the same angle. Depending on which book you look at or which teacher you ask or which website you're looking at, I've seen it as derivative of sine is negative cosine, derivative of cosine is negative sine, or as stated above, sine is derivative of cosine. (At least, I'm pretty sure I've seen all of those exprssions) Aren't all of these equal?

Secondly, since the sine and cosine values are 90 degrees out of phase with each other (meaning that they are perpendicular) and since the definition of derivative requires that the slope at the point of tangency be perpendicular and that lines (slopes in this case) are perpendicular if and only if the products of their slopes are negative 1 isn't it intuitive to deduce that the derivative of the sine is negative cosine and the derivative of the cosine is negative sine?

Bobby Brown
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    The derivative of sine is not negative cosine – graydad Sep 07 '15 at 20:18
  • $\frac{d}{dx}sinx=cosx$ – MathAdam Sep 07 '15 at 20:19
  • $D \sin x = \cos x$, $D \cos x = -\sin x$, and that's it. The other things you mention seem to come from a very bad book (if any). What's your question? Aren't all of what equal? Are derivative of sine negative cosine? Answer to the latter is: NO – skyking Sep 07 '15 at 20:21
  • I think that your memory is deceiving you, if you think you've heard anything other than $\frac d{dx}\sin x=\cos x$ and $\frac d{dx}\cos x=-\sin x$. Also, the graphs of $y=\sin x$ and $y=\cos x$ are not perpendicular. – Akiva Weinberger Sep 07 '15 at 20:25
  • Also, be careful: The derivative of, say, $\sin(\frac\pi4)$ is not $\cos(\frac\pi4)$. Why? Because that's a constant, and the derivative of any constant is $0$. (@peterwhy) – Akiva Weinberger Sep 07 '15 at 21:15
  • What do you mean by two functions being perpendicular? – mrf Sep 07 '15 at 21:29
  • "sine is derivative of cosine" is incorrect. – JRN Sep 08 '15 at 00:42

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The nice thing about trigonometric functions is that taking the derivative shifts the curve 90 degrees to the left. Sine and cosine each have 360 degree symmetry, and 180 degree reverse symmetry, so taking two derivatives always gets you the negative of the original function, and taking four gets you the original function back. Because $\cos$ is just $\sin$ shifted 90 degrees to the left, this gets you all the information you need.