The discussion revolves around calculating the number of smaller circles that can fit inside a larger circle while being tangent to its surface, specifically without using trigonometric functions. The conversation explores various mathematical approaches and reasoning related to geometry and circle packing.
Participants express differing views on the use of trigonometric functions, with some advocating for their use while others seek alternative methods. The discussion remains unresolved regarding the best approach to accurately determine the number of smaller circles.
Participants note that the accuracy of certain methods may depend on the assumptions made, such as the integer nature of the number of sides of polygons and the potential for fractional values in other contexts.
Thanks for the great work... but is that the closest that we can get ?.. i mean it's the right answer because N is equal 18 but what if in another case N has a fraction ... then by this method we can never know it's accurate valueanuttarasammyak said:A prescription is : Draw two lines from the center of large circle of radius R so that they are tangible to the small circle of radius r. How much is the angle between the lines?
With trigonometric, it is 2\ sin^{-1}\ \frac{r}{R-r}
Similar but another approach is, thinking N-regular polygon inside and tangent to the large circle, find N that satisfies
sin \frac{\pi}{N+1}< \frac{r}{R-r} < sin \frac{\pi}{N}.
Mostly we can replace the most LHS and RHS by their first term of Taylor expansion
\frac{\pi}{N+1}< \frac{r}{R-r} < \frac{\pi}{N}
where no trigonometric appear. For OP's case the middle term is 0.1737... so
N<18.08...<N+1
So N=18. I hope Taylor expansion approximation works at least for the OP's case.
anuttarasammyak said:N is a number of sides of polygon, so is an integer, with no fraction. If you are keen on accuracy, estimate second and higher terms of Taylor expansion or, as the straightest way , use trigonometry.