I need to mathematically prove that the center angle(s) (labeled as "A" in the photo below) approach what I believe to be 60 degrees (but never reach 60 degrees). We are given the values of all longer legs of each right triangle. Furthermore, the value of the length of each longer leg (corresponding to each right triangle) can be found by the formula 2^(n-1).
Finding angles with Trigonometry
General knowledge of sequences and series.
The Attempt at a Solution
First I noticed that triangle A1P1P2 is a 45-45-90 degree triangle, thus angle A1 is 45° and the length of each leg is 1.
Then I found the hypotenuse of triangle A1P1P2 using the Pythagorean Theorem; the hypotenuse is root 2 (as I depicted in the photo above).
To find the value of the angle P2A2P3 I used trigonometry:
Tanθ=2/root 2. θ≈54.7356. (as I depicted in the photo as "∠2=54.7356")
I again used the Pythagorean Theorem to find the hypotenuse of triangle A2P2P3; the hypotenuse is root 6 (as I depicted in the photo).
I repeated the steps above multiple times and found that ∠A sub n approaches 60°. Intuitively, this makes sense, but now I need to prove it using concepts from series and sequences.
Things I know:
⋅The hypotenuse of the first triangle, becomes the shortest leg of the next triangle
What I know is true about 30°-60°-90° triangles:
⋅The shortest leg is always ½ of its hypotenuse; alternatively, the hypotenuse is always 2 times longer than the shortest leg
⋅The longest leg is always equal to the shortest leg multiplied by root 3
Question I have:
I'm not sure how to incorporate the given formula for the lengths of the longer legs, 2^(n-1). I think I need to somehow come up with a series that calculates the ∠An for all values of n.
∈ tanθ=(2^(n-1)) / (shortest leg) Where "θ" is ∠A sub n.
Can anyone help me come up with a formula to replace "(shortest leg)" in the summation written above?
Can anyone shed some insights?
Lastly, my professor provided us with a hint:
∈4^k ; where the lower index is k=1, and the upper index is (n-1).
My professor also noted that the series equals (4-4^n) / (1-4).
Does this mean the series converges to (4-4^n) / (-3)?