In the video “Derivative of trigonometric functions”, specifically the example below:

I don’t understand why on the unit circle on the x-axis, the length between cos(x + Δx) and cos(x) is -Δ(cos x).

Could someone help by showing how this is derived?

A second question, later on in the video:

As Δx approaches 0, I don’t understand why the two angles circled start approximating each other.

Could someone explain the logic?

Thank you so much in advance.

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For your first question: as we go an infinitesimal (tiny) step in counter clockwise direction, the y axis value of the point on the unit circle will increase by Δ(sin x), and the x axis value will decrease by -Δ(cos x). Intuitively: because you are bound to the unit circle with radius 1, you are forced to “go up and left” if u are between 0 and 90 degrees (1st quadrant).

your second question: Think about what happens when you increase Δx by a huge amount. Imagine the orange triangle being so large, that it touches the y-axis with its left side (the sin(x) length arrow is touches the Δ(sin x) line). Now look at how long the sides Δ(sin x) and -Δ(cos x) are and compare it to the initial triangle in the video. You will see that the -Δ(cos x) is much longer than the Δ(sin x) line now right? Only for small Δx, the orange triangle will be the same as the intitial triangle (flipped by 90 degrees clockwise).

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I had so much frustration with this too and probably spent way too much time on it. The proof is from a trigonometric identity based on the **sin angle addition formula** (in this case Δx and x).

I like to have an intuitive picture but it will always fail without keeping the exceptions in mind. The most important part of this is that the arc length becomes “sufficiently small” and we can approximate Δx to be the length of the hypotenuse of the orange triangle. This allows it to approximate a similar (but not equal) triangle to the larger one. Similar triangles share the same angles which implies that the adjacent sides to each angle share a *change* in their relationship proportional to the said similar triangle.

There is a good example on Khanacademy of the proof Sin angle addition identity .

Note that the red colored 90 degree angles relate to the hypotenuses which are what the arrows are trying to demonstrate. The green 90 angle

*becomes closer* to size as the red 90 degree in the smaller triangle (as the angle Luis mentions approaches or “converges” with Δx)

Here is another attempt a diagram but if we rely on a picture without a proof, then diagram misleads us which is why it ends up looking like a paradox. Logic cannot be represented by a diagram sufficiently which is why calculus developed as it did (IMHO).

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I made another markup of my understanding of how this works (but I would like to know if I’m correct). It may be hard to read so I sincerely apologize. Also, I referenced Wikipedia for the definition of the arc length as I was trying to understand why *h* approaches Δx. Based on the formula of arc length where θ is in degrees ( π/180*r*θ). The radius is 1 on the unit circle and π/180 becomes 1 so Δx becomes an actual length of the hypotenuse (and not just degrees) Please see “Arcs of Circles” section. I welcome any corrections!

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Thank you both for your help and responses.

I also eventually understood the reasoning myself, from a 3blue1brown video and some comments under the video, specifically:

If anyone’s wondering what the justification is for the claim he makes at 15:39:

The base of the small triangle is perpendicular to the right side of the large triangle. The hypotenuse of the small triangle has a slope very close to the tangent of the circle at angle theta, and therefore is roughly perpendicular to the radius shown (the hypotenuse of the large triangle). Thus, the two angles of those two triangles that are touching are about the same. We also know they are both right triangles, so that’s two angles that match. There’s only one possible value left for the remaining angles (sum of interior angles of triangle = 180 degrees), so all the angles match, and therefore the two triangles are similar (well, mostly, but they get more similar for smaller values of d-theta).

I hope this helps other people as well.

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Thank You Wenxin. I was struggling to go forward without understanding this.

I am glad I checked the forum and the questions were exactly what I had in mind.

This helped.

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Oh that’s awesome, I’m happy to hear that this could help you!

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