Derivative of sinθ

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ray5450

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Mar 28, 2013
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I am not seeing something with this....

Assuming a unit circle with y=sinθ (with radius, θ at center, etc)...
If a small amount, dθ, is added to the angle, y also increases by dy:
y + dy = sin(θ + dθ)
dy = sin(θ + dθ) - y
If real values are used, say, θ=60°, dθ=1°...
y=sin(60°)=.8660
dy = sin(60° + 1°) - sin(60°)=.0086

The known derivative of sinθ is cosθ.
dy/dθ = cosθ (I think to operate with lengths, dθ is in radians?)
dy = cosθ * dθ
Using the same values:
dθ=1°=180/∏ rad
cosθ=.5
dy = .5 * 180/∏=28.64

Should these 2 routes obtain the same dy? If yes, what is my mistake? If no, why not?

Thanks.
 
You have the conversion from degrees to radians backwards. One degree is \(\displaystyle \frac{\pi}{180}\) radians.
 
OMG, I can't believe I did that.

Yes, that works.

I had copied and pasted the conversion from another post. Whatever that post was, it must have been converting the other way.

By the way, how did you create that fraction format?
 
Last edited:
ray5450 said:
...By the way, how did you create that fraction format?
Click to expand...

That's done using \(\displaystyle \LaTeX\), which is a markup language for creating beautiful math expressions, and is powered here by MathJax. To use it here at FMH, the easiest way is to wrap your code in [T E X][/T E X] tags (without the spaces...for some reason the noparse tags don't work here?).

For example, the code [T E X]\displaystyle \theta=\frac{\pi}{4}[/T E X]

produces:

\(\displaystyle \displaystyle \theta=\frac{\pi}{4}\)

If you want to see the code someone used for their expression, right-click the expression, and follow:

Show Math As ► TeX Command

And a window will popup showing you the code used. :D
 
There still seems to be a slight discrepancy.

0.5*pi/180 =0.008727

sin(61) - sin(60) = 0.00859

shouldn't they match exactly?
 
My guess is that the general proof of the known derivative cuts off some terms whose values are very small and approach zero.
 
hndalama said:
There still seems to be a slight discrepancy.

0.5*pi/180 =0.008727

sin(61) - sin(60) = 0.00859

shouldn't they match exactly?
Click to expand...
No. They should not match exactly.

The derivative is a limit. Informally,

\(\displaystyle \displaystyle \dfrac{dy}{dx} \approx \dfrac{\Delta y}{\Delta x} \implies \Delta y \approx \Delta x * \dfrac{dy}{dx}.\)

The smaller \(\displaystyle \Delta x\) the better the approximation.
 
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