A unit circle is made up of n wedges equivalent to the inner...

A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is $\sin \left(\frac{\pi}{n}\right) .$ The base of the outer triangle is $ B=\cos \left(\frac{\pi}{n}\right)+\sin \left(\frac{\pi}{n}\right) \tan \left(\frac{\pi}{n}\right) $ and the height is $H=B \sin \left(\frac{2 \pi}{n}\right)$. Use this information to argue that the area of a unit circle is equal to $\pi .$

Key Concepts

Unit Circle

A unit circle is a circle with a radius of one unit. It is a central object in trigonometry and geometry because it standardizes the relationship between angles and coordinates, making it a powerful tool for defining trigonometric functions and exploring circular properties.

Sector Decomposition (Wedge Approximation)

Dividing a circle into wedge-shaped sectors or 'wedges' is a method for approximating the area of a circle. By viewing each wedge as approximately a triangle, one can sum the areas of these nearly triangular pieces to approach the total area, with the approximation improving as the wedges become smaller.

Trigonometric Relationships

Trigonometric functions such as sine, cosine, and tangent are used to relate the angles of the sectors to linear measurements such as lengths, bases, and heights of the approximating triangles. These relationships are essential for expressing the dimensions of the wedges in terms of the central angles.

Limit Process and Convergence

Taking the limit as the number of wedges n approaches infinity is a key step in the argument. This process demonstrates that the sum of the areas of the increasingly narrow triangles converges to the exact area of the circle, illustrating the fundamental concept of convergence from discrete approximations to a continuous area.

Transcript

00:01 Okay, problem number 59.

00:04 We're going to try to argue that the unit circle has an area of pi.

00:09 And we're going to do that by just taking a portion of the unit circle, which i have outlined in black here, this little wedge.

00:19 And we're going to try to see if, if n of these, a, you know, an unknown amount of these wedges put together, can form an area of pi.

00:36 So first of all, we're just trying to represent one wedge, and we know the unit circle has a radius of one, and it has a height.

00:46 This little wedge is going to have a height of sign of 2 pi over n, so we can represent its area.

00:56 So we're going to do that.

01:00 The area of that little.

01:04 Actually, we're not going to represent the whole wedge.

01:06 We're just going to represent this.

01:09 We're going to cut off the outer part.

01:11 We're going to represent this little triangular part.

01:16 It's one half base, which is one, because it's a unit circle.

01:24 And then the height is sine of 2 pi over n.

01:37 Ok.

01:39 Now, what we're thinking is if this wedge gets smaller, and smaller and smaller and smaller, that this triangle is going to get closer and closer to the actual area of the wedge.

01:53 It'll just fit better inside of this wedge if this wedge gets smaller.

01:59 If i draw a really small wedge here, even smaller than that, and it's going to be really hard to tell the difference between the wedge and the triangle.

02:13 And if it gets even thinner, then it'll be even harder.

02:17 To tell.

02:18 So as the number of these wedges, as we have more and more wedges, then the area should get more and more accurate.

02:29 So that's what we're trying to figure out that, you know, the limit as the number of wedges gets very big, infinity, that this area, trying to figure out what this area will be.

02:50 I'm not going to go into a lesson of how to find limits or how to manipulate functions to find limits, but we can even use technology to find that the limit as n goes to infinity of this function is going to be pi.

03:08 Okay, well, that's interesting, that if we have an infinite amount of wedges of this size, actually, let me go back here.

03:21 It's not the limit of the area of one wedge.

03:27 It's the limit of the area of n of these wedges...

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