Transcription
If you've taken calculus, you probably showed the derivative of sine X is cosine X with some painful limit or identity. But there's a different, arguably better, argument, and this is nice and visual.
The way we define sine and cosine, cosine theta, sine theta is a point on the unit circle. The height, the Y value, is sine theta. The horizontal distance, the X, that's cosine theta, and it depends on the angle theta measured counterclockwise from the X axis.
Now, how fast the Y value, the Y part, the sine theta travels is its velocity. How fast it travels around the circle, that's its derivative. So, all we really have to do is find the velocity of this point as it travels around.
The point never leaves the circle, so it can't really move toward or away from the center. It can only turn, and that means its velocity is perpendicular to the radius. And on the unit circle, pushing the angle at rate one moves the point at a speed of exactly one. The velocity is a unit vector sitting at a right angle to the point cosine theta, sine theta pointing counterclockwise.
Rotate that point a quarter turn, and you get negative sine theta, cosine theta. Read off the height component. It's cosine theta, and so the derivative of sine theta is cosine theta, and for free you get the derivative of cosine theta is negative sine.
To be totally fair, the unit speed is essentially just the limit as H goes to zero of sine H over H equaling one, which is kind of what you probably used for the original derivation anyway, but I think this picture gives you something the algebra never did, and shows you why the answer is cosine. The height's speed is just the sideways position.