Basic geometry question re isosceles triangles

[michaelamarti]

I am reading "Mathematical Methods for Scientists and Engineers" by Donald McQuarrie. In his discussion of polar coordinates, he uses a geometric argument to derive the differential area element, which is of course rdrdθ. He shows an isosceles triangle whose two equal sides are r, and the angle that they make at the top of the triangle is delta(theta), where delta(theta) is small. From this he says it follows that the base of the triangle is equal to
r(delta)(theta).
However, I can not find this in my geometry book. Shouldn't the base be something like
r(sin)(delta(theta))?
I also ran into this same problem in an astrophysics book where the author used the very same argument for deriving the formula for the distance to a star by determining the parallax angle.
In both cases, are the authors using the "small angle approximation" that sin(theta) = theta?
Sorry to ask such a basic geometry question. I'm 61 years old, and I don't remember my high school geometry that well.

 

[Vargo]

Yes, they are using the small angle approximation that you refer to. Actually, the exact value would be 2rsin(delta/2), which comes out to r *delta using that approximation. These linear approximations are valid in integration theory because of the nature of the integral. It is defined as a limit of a sum as you partition the region of integration into small pieces. So on each small piece, the linear approximation is quite accurate. In the limit, (as the partition of the region gets finer and finer) the linear approximation is exactly correct.

 

[Mark44]

I am reading "Mathematical Methods for Scientists and Engineers" by Donald McQuarrie. In his discussion of polar coordinates, he uses a geometric argument to derive the differential area element, which is of course rdrdθ. He shows an isosceles triangle whose two equal sides are r, and the angle that they make at the top of the triangle is delta(theta), where delta(theta) is small. From this he says it follows that the base of the triangle is equal to
r(delta)(theta).
However, I can not find this in my geometry book. Shouldn't the base be something like
r(sin)(delta(theta))?

What you have written doesn't make much sense - sin by itself is as meaningful as √ by itself.

I don't have the book you cite, but I'm going to guess that he's approximating the area of part of a circular sector. This region is four-sided, with two of the sides being curved.

If you take what you're calling an isosceles triangle (actually it's a sector of a circle), and extend the two rays by Δr, you get the four-sided figure. It's roughly a rectangle, where one dimension is Δr and the other is rΔθ. If you multiply those, you get rΔθΔr, which is the same as rΔrΔθ.

The part that might be confusing you is the rΔθ. This is the length of the arc of a circle of radius r, that is subtended by an angle of measure Δθ. If the circle's radius is 1, the arc length will be 1Δθ or Δθ. If the circle's radius is 2, the arc length will be 2Δθ. In general, if the radius is r, the arc length is rΔθ.

I also ran into this same problem in an astrophysics book where the author used the very same argument for deriving the formula for the distance to a star by determining the parallax angle.
In both cases, are the authors using the "small angle approximation" that sin(theta) = theta?

Probably.

Sorry to ask such a basic geometry question. I'm 61 years old, and I don't remember my high school geometry that well.

It might not have been covered. When I took geometry (a few years before you did), most of the time we studied triangles, and didn't do much with other geometric figures.

 

[michaelamarti]

Thank you.
Holiday Greetings!

 

[CellCoree]

I can understand your confusion and it is perfectly normal to have questions about mathematical concepts, even if they seem basic. In this case, it seems that the authors are indeed using the small angle approximation, where sin(theta) is approximated to be equal to theta for small values of theta.

The reason for this approximation is that for small angles, the difference between the actual value of sin(theta) and the approximation is negligible. This makes calculations simpler and easier to understand. However, it is important to note that this approximation is only valid for small angles and may not hold true for larger angles.

In the context of isosceles triangles, the base of the triangle is indeed equal to r(delta)(theta) when using the small angle approximation. This is because for small angles, the base of the triangle can be approximated to be equal to the arc length along the circle, which is equal to r(theta). Therefore, the base can be expressed as r(delta)(theta).

I would recommend revisiting the concept of small angle approximation in your high school geometry book or consulting with a math teacher to clarify any doubts. It is always better to have a clear understanding of mathematical concepts before moving on to more advanced topics.

I hope this helps to clarify your doubts. Keep exploring and learning, and don't be afraid to ask questions. That is how science progresses. Good luck with your studies!