That area of the wedge to the bigger triangle? Well, it's clear that theĪrea of the salmon triangle is less than or equal Salmon-colored triangle which sits inside of this wedge and how do you compare Now, how would you compare the areas of this pink or this And so I can just write that down as the absolute value of the Our base, which is one, times our height, which is the absolute This is this entire are, that's going to be 1/2 times The area here is gonna beġ/2 times base times height. This larger triangle in this blue color, and this Value sign right over there 'cause we're talking about positive area. And if we wanted to make this work for thetas in the fourth quadrant, we could just write an absolute It has a radius one, so it'd be times the area of the circle, which would be pi times the radius square, the radius is one, so it's Two pis of the entire circle and we know the area of the circle. Way around the circle, it would be two pi radians, so this is theta over to So what fraction of the entireĬircle is this going to be? If I were to go all the The area of this wedge that I am highlighting I can just rewrite that as the absolute value of Going to be equal to 1/2 times our base, which is one, times our height, which is the absolute We know the height is the absolute value of the sine of theta and we know that the base is equal to one, so the area here is We know that the area of a triangle is 1/2 base times height. How can I express that area? Well, it's a triangle. And so let's think about the area of what I am shading in right over here. In this pie piece, this pie slice within the circle, so I can construct this triangle. So first, I'm gonna draw a triangle that sits in this wedge, So now that we've done, I'm gonna think about some triangles and their respective areas. Going to be a positive value for sitting here in the first quadrant but I want things to work in both the first and the fourth quadrant for the sake of our proof, so I'm just gonna putĪn absolute value here. So this just has length one, so the tangent of theta Triangle right over here, this is our angle theta in radians. Tangent of theta is equal to opposite over adjacent. What would tangent of theta be? Let me write it over here. Of a trigonometric function? Well, let's think about it. Now what about this blue line over here? Can I express that in terms It's the absolute value of the sine of theta. Quadrant, which will be useful, we can just insure that That also worked for thetas that end up in the fourth The unit circle definition of trig functions, the length of this line is going to be sine of theta. Would be the y-coordinate of where this radius So what does the length of this salmon-colored line represent? Well, the height of this line This is a unit circle, that we'll label it as such. Little bit of a geometric or trigonometric construction Going to do in this video is prove that the limit as theta approaches zero of sine of theta over theta is equal to one. If you really want to learn this type of thinking, a standard place to start is the book 'How to Solve It' by George Pólya. To draw the figure in the first place was fairly natural, because how else can we interpret the sine function if not by the unit circle? And once he had made the decision to compare areas, the proof was fairly straightforward algebra.Īs for how he came up with that idea, the answer is experience and intuition, the kind of intuition you build by writing a lot of proofs and studying a lot of different mathematical objects. The onus is very much put on the reader of the proof to slog through it word-by-word, and not on the writer to be clear.Īll that said, the only thing that Sal really pulled out of a hat was the idea to compare the different areas in the figure. The culture in the mathematics community dictates that once you've written a proof, you 'polish' it to make it as short and concise as possible. Before Common Core became widespread, it was common to see KA videos presenting the exact same topics in the exact same order as your own teacher.Ģ. grade-schoolers and high-schoolers, and so the videos on Khan Academy cater to these admittedly bad standards. The United States math curriculum places almost zero emphasis on proof writing or proof comprehension (and the things that pass for proofs in geometry are a joke). Before you continue, there are two unfortunate truths to keep in mind:ġ. Your objection is well-placed, and you're correct that proof-writing is a skill that is mostly improved by practice.
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