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06:46
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Crash Course: Physics
Trigonometry and Derivatives
06:46 - 08:30
In this video, the Crash Course expert, Shini, explains how trigonometry plays a role in derivatives. She displays the graphs of sin and cos and uses those to show how she arrived at the derivatives of each. She also defines the derivatives of each of the trigonomic measures (sin, cos, tan).
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Video Transcript
6:42
Now, there are a few more equations whose derivatives you should understand.
6:46
Trigonometry -- which we use to calculate the angles and sides of triangles -- is going
6:50
to come up a lot in physics, because we'll be using right angle triangles all the time.
6:54
So it's a good idea to know how to find the derivatives of sin(x) and cos(x).
6:58
Sine tells you that if you have a right angle triangle, and x is an angle in that triangle, then sin(x)
7:04
will be the (length of the side opposite, that angle), divided by the (hypotenuse).
7:07
Cosine does the same thing, just with the (side next to the angle) divided by the (hypotenuse).
7:12
So their graphs tell you what those ratios will be, depending on the angle.
7:16
We can actually try to guess the derivative of sin(x) just by looking at its graph.
7:20
You can see that the curve has turning points every so often, at x = -90 degrees, x = 90
7:25
degrees, and so on -- repeating every 180 degrees.
7:29
Meaning, at those points, the equations aren't changing at all -- so the derivative at these
7:35
turning points is also going to be exactly zero.
7:37
Let's pull up another graph where we'll plot the derivative, and put little dots where
7:40
we know it'll be zero.
7:42
Now, what's happening between those turning points? Well, from -270 to -90 degrees, sin(x)
7:49
is decreasing.
7:49
In other words, its change -- and therefore its derivative -- must be negative.
7:54
Then, from -90 to 90 degrees, sin(x) is increasing -- so it'll have a positive derivative. And so on...
8:01
There are actually a lot more clues in this graph to help us find the derivative, but
8:05
we already know enough to make a decent guess.
8:07
If we smoothly connect the dots on the graph of our derivative, keeping in mind where the
8:11
curve should be positive and where it should be negative … hey, this derivative is looking
8:16
a whole lot like the graph of cos(x)!
8:18
That’s because it is. The derivative of sine is just cosine, and that is going to
8:22
come up a LOT.
8:24
So will these, which you can work out on your own by repeating what we just did with the
8:28
graphs of sin(x) and cos(x).
