Hey for this course, in 3:30, I don’t understand two things:
- Why one edge is - delta cos(X)
- Why another edge is delta sin(X)
Anyone can help?
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Here is the screenshot:
The first thing is to realize that the definitions are:
\Delta sin(x) = sin(x + \Delta x) - sin(x)
\Delta cos(x) = cos(x + \Delta x) - cos(x)
Now look at the diagram. The length of the bottom edge of the small orange triangle is:
cos(x) - cos(x + \Delta x)
So that is equal to
-\Delta cos(x)
Now apply the same logic to the left edge of the orange triangle. The length is:
sin(x + \Delta x) - sin(x)
I’m also confused on this: subtracting the leg cos(x+Δx) from cos(x)
cos(x)-cos(x+Δx)
cos(x)-(cos(x)+cos(Δx))
cos(x)-cos(x)-cos(Δx))
-cos(Δx)
?
The trig functions are not linear functions, right? So we have:
cos(x + \Delta x) \neq cos(x) + cos(\Delta x)
Have you taken trig? You should remember the formulas:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
So apply that second formula:
cos(x + \Delta x) = cos(x)cos(\Delta x) - sin(x)sin(\Delta x)
Then we consider what happens when \Delta x \rightarrow 0.
Or maybe I’m missing your point. As I said in my original post on this thread, this is just a definition:
\Delta cos(x) = cos(x + \Delta x) - cos(x)
The “delta” or “change in” cos(x) is that value on the RHS.
\Delta cos(x) is not the same thing as cos(\Delta x), right?
I was a bit confused by what was stated above and frankly spent too much time on this; for anyone looking in the future, I found the videos on this webpage helpful Khan Academy, hopefully it helps someone else also